Step | Hyp | Ref
| Expression |
1 | | txcnp.4 |
. . . . . . 7
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
2 | | txcnp.5 |
. . . . . . 7
⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) |
3 | | txcnp.8 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷)) |
4 | | cnpf2 20864 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌) |
5 | 1, 2, 3, 4 | syl3anc 1318 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌) |
6 | | eqid 2610 |
. . . . . . 7
⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴) |
7 | 6 | fmpt 6289 |
. . . . . 6
⊢
(∀𝑥 ∈
𝑋 𝐴 ∈ 𝑌 ↔ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌) |
8 | 5, 7 | sylibr 223 |
. . . . 5
⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌) |
9 | 8 | r19.21bi 2916 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) |
10 | | txcnp.6 |
. . . . . . 7
⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) |
11 | | txcnp.9 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷)) |
12 | | cnpf2 20864 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍) |
13 | 1, 10, 11, 12 | syl3anc 1318 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍) |
14 | | eqid 2610 |
. . . . . . 7
⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵) |
15 | 14 | fmpt 6289 |
. . . . . 6
⊢
(∀𝑥 ∈
𝑋 𝐵 ∈ 𝑍 ↔ (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍) |
16 | 13, 15 | sylibr 223 |
. . . . 5
⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑍) |
17 | 16 | r19.21bi 2916 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑍) |
18 | | opelxpi 5072 |
. . . 4
⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍) → 〈𝐴, 𝐵〉 ∈ (𝑌 × 𝑍)) |
19 | 9, 17, 18 | syl2anc 691 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ (𝑌 × 𝑍)) |
20 | | eqid 2610 |
. . 3
⊢ (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) = (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) |
21 | 19, 20 | fmptd 6292 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉):𝑋⟶(𝑌 × 𝑍)) |
22 | | txcnp.7 |
. . . . . . . . 9
⊢ (𝜑 → 𝐷 ∈ 𝑋) |
23 | | simpr 476 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
24 | | opex 4859 |
. . . . . . . . . . . 12
⊢
〈𝐴, 𝐵〉 ∈ V |
25 | 20 | fvmpt2 6200 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ 𝑋 ∧ 〈𝐴, 𝐵〉 ∈ V) → ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑥) = 〈𝐴, 𝐵〉) |
26 | 23, 24, 25 | sylancl 693 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑥) = 〈𝐴, 𝐵〉) |
27 | 6 | fvmpt2 6200 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴) |
28 | 23, 9, 27 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴) |
29 | 14 | fvmpt2 6200 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ 𝑋 ∧ 𝐵 ∈ 𝑍) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = 𝐵) |
30 | 23, 17, 29 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = 𝐵) |
31 | 28, 30 | opeq12d 4348 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)〉 = 〈𝐴, 𝐵〉) |
32 | 26, 31 | eqtr4d 2647 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑥) = 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)〉) |
33 | 32 | ralrimiva 2949 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑥) = 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)〉) |
34 | | nffvmpt1 6111 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) |
35 | | nffvmpt1 6111 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) |
36 | | nffvmpt1 6111 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) |
37 | 35, 36 | nfop 4356 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)〉 |
38 | 34, 37 | nfeq 2762 |
. . . . . . . . . 10
⊢
Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) = 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)〉 |
39 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝐷 → ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷)) |
40 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝐷 → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷)) |
41 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝐷 → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)) |
42 | 40, 41 | opeq12d 4348 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝐷 → 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)〉 = 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)〉) |
43 | 39, 42 | eqeq12d 2625 |
. . . . . . . . . 10
⊢ (𝑥 = 𝐷 → (((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑥) = 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)〉 ↔ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) = 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)〉)) |
44 | 38, 43 | rspc 3276 |
. . . . . . . . 9
⊢ (𝐷 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑥) = 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)〉 → ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) = 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)〉)) |
45 | 22, 33, 44 | sylc 63 |
. . . . . . . 8
⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) = 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)〉) |
46 | 45 | eleq1d 2672 |
. . . . . . 7
⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) ∈ (𝑣 × 𝑤) ↔ 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)〉 ∈ (𝑣 × 𝑤))) |
47 | 46 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) → (((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) ∈ (𝑣 × 𝑤) ↔ 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)〉 ∈ (𝑣 × 𝑤))) |
48 | 3 | ad2antrr 758 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷)) |
49 | | simplrl 796 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → 𝑣 ∈ 𝐾) |
50 | | simprl 790 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣) |
51 | | cnpimaex 20870 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷) ∧ 𝑣 ∈ 𝐾 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣) → ∃𝑟 ∈ 𝐽 (𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣)) |
52 | 48, 49, 50, 51 | syl3anc 1318 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → ∃𝑟 ∈ 𝐽 (𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣)) |
53 | 11 | ad2antrr 758 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷)) |
54 | | simplrr 797 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → 𝑤 ∈ 𝐿) |
55 | | simprr 792 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤) |
56 | | cnpimaex 20870 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷) ∧ 𝑤 ∈ 𝐿 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤) → ∃𝑠 ∈ 𝐽 (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)) |
57 | 53, 54, 55, 56 | syl3anc 1318 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → ∃𝑠 ∈ 𝐽 (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)) |
58 | 52, 57 | jca 553 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) → (∃𝑟 ∈ 𝐽 (𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ ∃𝑠 ∈ 𝐽 (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤))) |
59 | 58 | ex 449 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) → ((((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤) → (∃𝑟 ∈ 𝐽 (𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ ∃𝑠 ∈ 𝐽 (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)))) |
60 | | opelxp 5070 |
. . . . . . 7
⊢
(〈((𝑥 ∈
𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)〉 ∈ (𝑣 × 𝑤) ↔ (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷) ∈ 𝑤)) |
61 | | reeanv 3086 |
. . . . . . 7
⊢
(∃𝑟 ∈
𝐽 ∃𝑠 ∈ 𝐽 ((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)) ↔ (∃𝑟 ∈ 𝐽 (𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ ∃𝑠 ∈ 𝐽 (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤))) |
62 | 59, 60, 61 | 3imtr4g 284 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) → (〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝐷), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝐷)〉 ∈ (𝑣 × 𝑤) → ∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐽 ((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)))) |
63 | 47, 62 | sylbid 229 |
. . . . 5
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) → (((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐽 ((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)))) |
64 | | an4 861 |
. . . . . . . . . . 11
⊢ (((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)) ↔ ((𝐷 ∈ 𝑟 ∧ 𝐷 ∈ 𝑠) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤))) |
65 | | elin 3758 |
. . . . . . . . . . . . . 14
⊢ (𝐷 ∈ (𝑟 ∩ 𝑠) ↔ (𝐷 ∈ 𝑟 ∧ 𝐷 ∈ 𝑠)) |
66 | 65 | biimpri 217 |
. . . . . . . . . . . . 13
⊢ ((𝐷 ∈ 𝑟 ∧ 𝐷 ∈ 𝑠) → 𝐷 ∈ (𝑟 ∩ 𝑠)) |
67 | 66 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → ((𝐷 ∈ 𝑟 ∧ 𝐷 ∈ 𝑠) → 𝐷 ∈ (𝑟 ∩ 𝑠))) |
68 | | simpl 472 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽) → 𝑟 ∈ 𝐽) |
69 | | toponss 20544 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑟 ∈ 𝐽) → 𝑟 ⊆ 𝑋) |
70 | 1, 68, 69 | syl2an 493 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → 𝑟 ⊆ 𝑋) |
71 | | ssinss1 3803 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑟 ⊆ 𝑋 → (𝑟 ∩ 𝑠) ⊆ 𝑋) |
72 | 71 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → (𝑟 ∩ 𝑠) ⊆ 𝑋) |
73 | 72 | sselda 3568 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → 𝑡 ∈ 𝑋) |
74 | 33 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → ∀𝑥 ∈ 𝑋 ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑥) = 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)〉) |
75 | | nffvmpt1 6111 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑡) |
76 | | nffvmpt1 6111 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡) |
77 | | nffvmpt1 6111 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡) |
78 | 76, 77 | nfop 4356 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑥〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)〉 |
79 | 75, 78 | nfeq 2762 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑥((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑡) = 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)〉 |
80 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑡 → ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑡)) |
81 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑡 → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡)) |
82 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑡 → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)) |
83 | 81, 82 | opeq12d 4348 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑡 → 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)〉 = 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)〉) |
84 | 80, 83 | eqeq12d 2625 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 𝑡 → (((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑥) = 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)〉 ↔ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑡) = 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)〉)) |
85 | 79, 84 | rspc 3276 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑥) = 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑥)〉 → ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑡) = 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)〉)) |
86 | 73, 74, 85 | sylc 63 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑡) = 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)〉) |
87 | | inss1 3795 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑟 ∩ 𝑠) ⊆ 𝑟 |
88 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → 𝑡 ∈ (𝑟 ∩ 𝑠)) |
89 | 87, 88 | sseldi 3566 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → 𝑡 ∈ 𝑟) |
90 | 5 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌) |
91 | | ffun 5961 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌 → Fun (𝑥 ∈ 𝑋 ↦ 𝐴)) |
92 | 90, 91 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → Fun (𝑥 ∈ 𝑋 ↦ 𝐴)) |
93 | 72 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑟 ∩ 𝑠) ⊆ 𝑋) |
94 | | fdm 5964 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌 → dom (𝑥 ∈ 𝑋 ↦ 𝐴) = 𝑋) |
95 | 90, 94 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → dom (𝑥 ∈ 𝑋 ↦ 𝐴) = 𝑋) |
96 | 93, 95 | sseqtr4d 3605 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑟 ∩ 𝑠) ⊆ dom (𝑥 ∈ 𝑋 ↦ 𝐴)) |
97 | 96, 88 | sseldd 3569 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → 𝑡 ∈ dom (𝑥 ∈ 𝑋 ↦ 𝐴)) |
98 | | funfvima 6396 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((Fun
(𝑥 ∈ 𝑋 ↦ 𝐴) ∧ 𝑡 ∈ dom (𝑥 ∈ 𝑋 ↦ 𝐴)) → (𝑡 ∈ 𝑟 → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟))) |
99 | 92, 97, 98 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑡 ∈ 𝑟 → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟))) |
100 | 89, 99 | mpd 15 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟)) |
101 | | inss2 3796 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑟 ∩ 𝑠) ⊆ 𝑠 |
102 | 101, 88 | sseldi 3566 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → 𝑡 ∈ 𝑠) |
103 | 13 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍) |
104 | | ffun 5961 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍 → Fun (𝑥 ∈ 𝑋 ↦ 𝐵)) |
105 | 103, 104 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → Fun (𝑥 ∈ 𝑋 ↦ 𝐵)) |
106 | | fdm 5964 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍 → dom (𝑥 ∈ 𝑋 ↦ 𝐵) = 𝑋) |
107 | 103, 106 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → dom (𝑥 ∈ 𝑋 ↦ 𝐵) = 𝑋) |
108 | 93, 107 | sseqtr4d 3605 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑟 ∩ 𝑠) ⊆ dom (𝑥 ∈ 𝑋 ↦ 𝐵)) |
109 | 108, 88 | sseldd 3569 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → 𝑡 ∈ dom (𝑥 ∈ 𝑋 ↦ 𝐵)) |
110 | | funfvima 6396 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((Fun
(𝑥 ∈ 𝑋 ↦ 𝐵) ∧ 𝑡 ∈ dom (𝑥 ∈ 𝑋 ↦ 𝐵)) → (𝑡 ∈ 𝑠 → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠))) |
111 | 105, 109,
110 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → (𝑡 ∈ 𝑠 → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠))) |
112 | 102, 111 | mpd 15 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)) |
113 | | opelxpi 5072 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡) ∈ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)) → 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)〉 ∈ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠))) |
114 | 100, 112,
113 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → 〈((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑡), ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑡)〉 ∈ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠))) |
115 | 86, 114 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑟 ⊆ 𝑋) ∧ 𝑡 ∈ (𝑟 ∩ 𝑠)) → ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑡) ∈ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠))) |
116 | 115 | ralrimiva 2949 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → ∀𝑡 ∈ (𝑟 ∩ 𝑠)((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑡) ∈ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠))) |
117 | | ffun 5961 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉):𝑋⟶(𝑌 × 𝑍) → Fun (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)) |
118 | 21, 117 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → Fun (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)) |
119 | 118 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → Fun (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)) |
120 | | fdm 5964 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉):𝑋⟶(𝑌 × 𝑍) → dom (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) = 𝑋) |
121 | 21, 120 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) = 𝑋) |
122 | 121 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → dom (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) = 𝑋) |
123 | 72, 122 | sseqtr4d 3605 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → (𝑟 ∩ 𝑠) ⊆ dom (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)) |
124 | | funimass4 6157 |
. . . . . . . . . . . . . . . . 17
⊢ ((Fun
(𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) ∧ (𝑟 ∩ 𝑠) ⊆ dom (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)) → (((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ (𝑟 ∩ 𝑠)) ⊆ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)) ↔ ∀𝑡 ∈ (𝑟 ∩ 𝑠)((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑡) ∈ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)))) |
125 | 119, 123,
124 | syl2anc 691 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → (((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ (𝑟 ∩ 𝑠)) ⊆ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)) ↔ ∀𝑡 ∈ (𝑟 ∩ 𝑠)((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝑡) ∈ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)))) |
126 | 116, 125 | mpbird 246 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑟 ⊆ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ (𝑟 ∩ 𝑠)) ⊆ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠))) |
127 | 70, 126 | syldan 486 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ (𝑟 ∩ 𝑠)) ⊆ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠))) |
128 | 127 | adantlr 747 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ (𝑟 ∩ 𝑠)) ⊆ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠))) |
129 | | xpss12 5148 |
. . . . . . . . . . . . 13
⊢ ((((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤) → (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)) ⊆ (𝑣 × 𝑤)) |
130 | | sstr2 3575 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ (𝑟 ∩ 𝑠)) ⊆ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)) → ((((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) × ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠)) ⊆ (𝑣 × 𝑤) → ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤))) |
131 | 128, 129,
130 | syl2im 39 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → ((((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤) → ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤))) |
132 | 67, 131 | anim12d 584 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → (((𝐷 ∈ 𝑟 ∧ 𝐷 ∈ 𝑠) ∧ (((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)) → (𝐷 ∈ (𝑟 ∩ 𝑠) ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤)))) |
133 | 64, 132 | syl5bi 231 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → (((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)) → (𝐷 ∈ (𝑟 ∩ 𝑠) ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤)))) |
134 | | topontop 20541 |
. . . . . . . . . . . . 13
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) |
135 | 1, 134 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐽 ∈ Top) |
136 | | inopn 20529 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ 𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽) → (𝑟 ∩ 𝑠) ∈ 𝐽) |
137 | 136 | 3expb 1258 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → (𝑟 ∩ 𝑠) ∈ 𝐽) |
138 | 135, 137 | sylan 487 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → (𝑟 ∩ 𝑠) ∈ 𝐽) |
139 | 138 | adantlr 747 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → (𝑟 ∩ 𝑠) ∈ 𝐽) |
140 | 133, 139 | jctild 564 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) ∧ (𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽)) → (((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)) → ((𝑟 ∩ 𝑠) ∈ 𝐽 ∧ (𝐷 ∈ (𝑟 ∩ 𝑠) ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤))))) |
141 | 140 | expimpd 627 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) → (((𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽) ∧ ((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤))) → ((𝑟 ∩ 𝑠) ∈ 𝐽 ∧ (𝐷 ∈ (𝑟 ∩ 𝑠) ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤))))) |
142 | | eleq2 2677 |
. . . . . . . . . 10
⊢ (𝑧 = (𝑟 ∩ 𝑠) → (𝐷 ∈ 𝑧 ↔ 𝐷 ∈ (𝑟 ∩ 𝑠))) |
143 | | imaeq2 5381 |
. . . . . . . . . . 11
⊢ (𝑧 = (𝑟 ∩ 𝑠) → ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) = ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ (𝑟 ∩ 𝑠))) |
144 | 143 | sseq1d 3595 |
. . . . . . . . . 10
⊢ (𝑧 = (𝑟 ∩ 𝑠) → (((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ (𝑣 × 𝑤) ↔ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤))) |
145 | 142, 144 | anbi12d 743 |
. . . . . . . . 9
⊢ (𝑧 = (𝑟 ∩ 𝑠) → ((𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ (𝑣 × 𝑤)) ↔ (𝐷 ∈ (𝑟 ∩ 𝑠) ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤)))) |
146 | 145 | rspcev 3282 |
. . . . . . . 8
⊢ (((𝑟 ∩ 𝑠) ∈ 𝐽 ∧ (𝐷 ∈ (𝑟 ∩ 𝑠) ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ (𝑟 ∩ 𝑠)) ⊆ (𝑣 × 𝑤))) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ (𝑣 × 𝑤))) |
147 | 141, 146 | syl6 34 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) → (((𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽) ∧ ((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤))) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ (𝑣 × 𝑤)))) |
148 | 147 | expd 451 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) → ((𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽) → (((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ (𝑣 × 𝑤))))) |
149 | 148 | rexlimdvv 3019 |
. . . . 5
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) → (∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐽 ((𝐷 ∈ 𝑟 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷 ∈ 𝑠 ∧ ((𝑥 ∈ 𝑋 ↦ 𝐵) “ 𝑠) ⊆ 𝑤)) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ (𝑣 × 𝑤)))) |
150 | 63, 149 | syld 46 |
. . . 4
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐾 ∧ 𝑤 ∈ 𝐿)) → (((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ (𝑣 × 𝑤)))) |
151 | 150 | ralrimivva 2954 |
. . 3
⊢ (𝜑 → ∀𝑣 ∈ 𝐾 ∀𝑤 ∈ 𝐿 (((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ (𝑣 × 𝑤)))) |
152 | | vex 3176 |
. . . . . 6
⊢ 𝑣 ∈ V |
153 | | vex 3176 |
. . . . . 6
⊢ 𝑤 ∈ V |
154 | 152, 153 | xpex 6860 |
. . . . 5
⊢ (𝑣 × 𝑤) ∈ V |
155 | 154 | rgen2w 2909 |
. . . 4
⊢
∀𝑣 ∈
𝐾 ∀𝑤 ∈ 𝐿 (𝑣 × 𝑤) ∈ V |
156 | | eqid 2610 |
. . . . 5
⊢ (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤)) = (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤)) |
157 | | eleq2 2677 |
. . . . . 6
⊢ (𝑦 = (𝑣 × 𝑤) → (((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) ∈ 𝑦 ↔ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) ∈ (𝑣 × 𝑤))) |
158 | | sseq2 3590 |
. . . . . . . 8
⊢ (𝑦 = (𝑣 × 𝑤) → (((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ 𝑦 ↔ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ (𝑣 × 𝑤))) |
159 | 158 | anbi2d 736 |
. . . . . . 7
⊢ (𝑦 = (𝑣 × 𝑤) → ((𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ 𝑦) ↔ (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ (𝑣 × 𝑤)))) |
160 | 159 | rexbidv 3034 |
. . . . . 6
⊢ (𝑦 = (𝑣 × 𝑤) → (∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ 𝑦) ↔ ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ (𝑣 × 𝑤)))) |
161 | 157, 160 | imbi12d 333 |
. . . . 5
⊢ (𝑦 = (𝑣 × 𝑤) → ((((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) ∈ 𝑦 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ 𝑦)) ↔ (((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ (𝑣 × 𝑤))))) |
162 | 156, 161 | ralrnmpt2 6673 |
. . . 4
⊢
(∀𝑣 ∈
𝐾 ∀𝑤 ∈ 𝐿 (𝑣 × 𝑤) ∈ V → (∀𝑦 ∈ ran (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤))(((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) ∈ 𝑦 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ 𝑦)) ↔ ∀𝑣 ∈ 𝐾 ∀𝑤 ∈ 𝐿 (((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ (𝑣 × 𝑤))))) |
163 | 155, 162 | ax-mp 5 |
. . 3
⊢
(∀𝑦 ∈
ran (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤))(((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) ∈ 𝑦 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ 𝑦)) ↔ ∀𝑣 ∈ 𝐾 ∀𝑤 ∈ 𝐿 (((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ (𝑣 × 𝑤)))) |
164 | 151, 163 | sylibr 223 |
. 2
⊢ (𝜑 → ∀𝑦 ∈ ran (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤))(((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) ∈ 𝑦 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ 𝑦))) |
165 | | topontop 20541 |
. . . . 5
⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top) |
166 | 2, 165 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐾 ∈ Top) |
167 | | topontop 20541 |
. . . . 5
⊢ (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top) |
168 | 10, 167 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐿 ∈ Top) |
169 | | eqid 2610 |
. . . . 5
⊢ ran
(𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤)) = ran (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤)) |
170 | 169 | txval 21177 |
. . . 4
⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐾 ×t 𝐿) = (topGen‘ran (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤)))) |
171 | 166, 168,
170 | syl2anc 691 |
. . 3
⊢ (𝜑 → (𝐾 ×t 𝐿) = (topGen‘ran (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤)))) |
172 | | txtopon 21204 |
. . . 4
⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍))) |
173 | 2, 10, 172 | syl2anc 691 |
. . 3
⊢ (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍))) |
174 | 1, 171, 173, 22 | tgcnp 20867 |
. 2
⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) ∈ ((𝐽 CnP (𝐾 ×t 𝐿))‘𝐷) ↔ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉):𝑋⟶(𝑌 × 𝑍) ∧ ∀𝑦 ∈ ran (𝑣 ∈ 𝐾, 𝑤 ∈ 𝐿 ↦ (𝑣 × 𝑤))(((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉)‘𝐷) ∈ 𝑦 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) “ 𝑧) ⊆ 𝑦))))) |
175 | 21, 164, 174 | mpbir2and 959 |
1
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) ∈ ((𝐽 CnP (𝐾 ×t 𝐿))‘𝐷)) |