Step | Hyp | Ref
| Expression |
1 | | simpl3 1059 |
. . . . . . 7
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝐹:𝑌⟶𝑋) |
2 | | cnvimass 5404 |
. . . . . . . 8
⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹 |
3 | | fdm 5964 |
. . . . . . . 8
⊢ (𝐹:𝑌⟶𝑋 → dom 𝐹 = 𝑌) |
4 | 2, 3 | syl5sseq 3616 |
. . . . . . 7
⊢ (𝐹:𝑌⟶𝑋 → (◡𝐹 “ 𝑥) ⊆ 𝑌) |
5 | 1, 4 | syl 17 |
. . . . . 6
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (◡𝐹 “ 𝑥) ⊆ 𝑌) |
6 | | simpl1 1057 |
. . . . . . 7
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝑌 ∈ 𝐴) |
7 | | elpw2g 4754 |
. . . . . . 7
⊢ (𝑌 ∈ 𝐴 → ((◡𝐹 “ 𝑥) ∈ 𝒫 𝑌 ↔ (◡𝐹 “ 𝑥) ⊆ 𝑌)) |
8 | 6, 7 | syl 17 |
. . . . . 6
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ((◡𝐹 “ 𝑥) ∈ 𝒫 𝑌 ↔ (◡𝐹 “ 𝑥) ⊆ 𝑌)) |
9 | 5, 8 | mpbird 246 |
. . . . 5
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (◡𝐹 “ 𝑥) ∈ 𝒫 𝑌) |
10 | 9 | adantr 480 |
. . . 4
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) → (◡𝐹 “ 𝑥) ∈ 𝒫 𝑌) |
11 | | eqid 2610 |
. . . 4
⊢ (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) = (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) |
12 | 10, 11 | fmptd 6292 |
. . 3
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)):𝐿⟶𝒫 𝑌) |
13 | | frn 5966 |
. . 3
⊢ ((𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)):𝐿⟶𝒫 𝑌 → ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ⊆ 𝒫 𝑌) |
14 | 12, 13 | syl 17 |
. 2
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ⊆ 𝒫 𝑌) |
15 | | filtop 21469 |
. . . . . . . 8
⊢ (𝐿 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐿) |
16 | 15 | 3ad2ant2 1076 |
. . . . . . 7
⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → 𝑋 ∈ 𝐿) |
17 | 16 | adantr 480 |
. . . . . 6
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝑋 ∈ 𝐿) |
18 | | fimacnv 6255 |
. . . . . . . . 9
⊢ (𝐹:𝑌⟶𝑋 → (◡𝐹 “ 𝑋) = 𝑌) |
19 | 18 | eqcomd 2616 |
. . . . . . . 8
⊢ (𝐹:𝑌⟶𝑋 → 𝑌 = (◡𝐹 “ 𝑋)) |
20 | 19 | 3ad2ant3 1077 |
. . . . . . 7
⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → 𝑌 = (◡𝐹 “ 𝑋)) |
21 | 20 | adantr 480 |
. . . . . 6
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝑌 = (◡𝐹 “ 𝑋)) |
22 | | imaeq2 5381 |
. . . . . . . 8
⊢ (𝑥 = 𝑋 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑋)) |
23 | 22 | eqeq2d 2620 |
. . . . . . 7
⊢ (𝑥 = 𝑋 → (𝑌 = (◡𝐹 “ 𝑥) ↔ 𝑌 = (◡𝐹 “ 𝑋))) |
24 | 23 | rspcev 3282 |
. . . . . 6
⊢ ((𝑋 ∈ 𝐿 ∧ 𝑌 = (◡𝐹 “ 𝑋)) → ∃𝑥 ∈ 𝐿 𝑌 = (◡𝐹 “ 𝑥)) |
25 | 17, 21, 24 | syl2anc 691 |
. . . . 5
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ∃𝑥 ∈ 𝐿 𝑌 = (◡𝐹 “ 𝑥)) |
26 | 11 | elrnmpt 5293 |
. . . . . . 7
⊢ (𝑌 ∈ 𝐴 → (𝑌 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑌 = (◡𝐹 “ 𝑥))) |
27 | 26 | 3ad2ant1 1075 |
. . . . . 6
⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑌 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑌 = (◡𝐹 “ 𝑥))) |
28 | 27 | adantr 480 |
. . . . 5
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (𝑌 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑌 = (◡𝐹 “ 𝑥))) |
29 | 25, 28 | mpbird 246 |
. . . 4
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝑌 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) |
30 | | ne0i 3880 |
. . . 4
⊢ (𝑌 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) → ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ≠ ∅) |
31 | 29, 30 | syl 17 |
. . 3
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ≠ ∅) |
32 | | 0nelfil 21463 |
. . . . . . 7
⊢ (𝐿 ∈ (Fil‘𝑋) → ¬ ∅ ∈
𝐿) |
33 | 32 | 3ad2ant2 1076 |
. . . . . 6
⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → ¬ ∅ ∈ 𝐿) |
34 | 33 | adantr 480 |
. . . . 5
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ¬ ∅ ∈ 𝐿) |
35 | | 0ex 4718 |
. . . . . . 7
⊢ ∅
∈ V |
36 | 11 | elrnmpt 5293 |
. . . . . . 7
⊢ (∅
∈ V → (∅ ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 ∅ = (◡𝐹 “ 𝑥))) |
37 | 35, 36 | ax-mp 5 |
. . . . . 6
⊢ (∅
∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 ∅ = (◡𝐹 “ 𝑥)) |
38 | | ffn 5958 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐹:𝑌⟶𝑋 → 𝐹 Fn 𝑌) |
39 | | fvelrnb 6153 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐹 Fn 𝑌 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = 𝑦)) |
40 | 38, 39 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐹:𝑌⟶𝑋 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = 𝑦)) |
41 | 40 | 3ad2ant3 1077 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = 𝑦)) |
42 | 41 | ad2antrr 758 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑥)) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = 𝑦)) |
43 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹‘𝑧) = 𝑦 → ((𝐹‘𝑧) ∈ 𝑥 ↔ 𝑦 ∈ 𝑥)) |
44 | 43 | biimparc 503 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑦 ∈ 𝑥 ∧ (𝐹‘𝑧) = 𝑦) → (𝐹‘𝑧) ∈ 𝑥) |
45 | 44 | ad2ant2l 778 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑥) ∧ (𝑧 ∈ 𝑌 ∧ (𝐹‘𝑧) = 𝑦)) → (𝐹‘𝑧) ∈ 𝑥) |
46 | 45 | adantll 746 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑧 ∈ 𝑌 ∧ (𝐹‘𝑧) = 𝑦)) → (𝐹‘𝑧) ∈ 𝑥) |
47 | | ffun 5961 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐹:𝑌⟶𝑋 → Fun 𝐹) |
48 | 47 | 3ad2ant3 1077 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → Fun 𝐹) |
49 | 48 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑧 ∈ 𝑌 ∧ (𝐹‘𝑧) = 𝑦)) → Fun 𝐹) |
50 | 3 | eleq2d 2673 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐹:𝑌⟶𝑋 → (𝑧 ∈ dom 𝐹 ↔ 𝑧 ∈ 𝑌)) |
51 | 50 | biimpar 501 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝑧 ∈ 𝑌) → 𝑧 ∈ dom 𝐹) |
52 | 51 | 3ad2antl3 1218 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑧 ∈ 𝑌) → 𝑧 ∈ dom 𝐹) |
53 | 52 | adantlr 747 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑧 ∈ 𝑌) → 𝑧 ∈ dom 𝐹) |
54 | 53 | ad2ant2r 779 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑧 ∈ 𝑌 ∧ (𝐹‘𝑧) = 𝑦)) → 𝑧 ∈ dom 𝐹) |
55 | | fvimacnv 6240 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((Fun
𝐹 ∧ 𝑧 ∈ dom 𝐹) → ((𝐹‘𝑧) ∈ 𝑥 ↔ 𝑧 ∈ (◡𝐹 “ 𝑥))) |
56 | 49, 54, 55 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑧 ∈ 𝑌 ∧ (𝐹‘𝑧) = 𝑦)) → ((𝐹‘𝑧) ∈ 𝑥 ↔ 𝑧 ∈ (◡𝐹 “ 𝑥))) |
57 | 46, 56 | mpbid 221 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑧 ∈ 𝑌 ∧ (𝐹‘𝑧) = 𝑦)) → 𝑧 ∈ (◡𝐹 “ 𝑥)) |
58 | | n0i 3879 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ (◡𝐹 “ 𝑥) → ¬ (◡𝐹 “ 𝑥) = ∅) |
59 | | eqcom 2617 |
. . . . . . . . . . . . . . . . . 18
⊢ ((◡𝐹 “ 𝑥) = ∅ ↔ ∅ = (◡𝐹 “ 𝑥)) |
60 | 58, 59 | sylnib 317 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 ∈ (◡𝐹 “ 𝑥) → ¬ ∅ = (◡𝐹 “ 𝑥)) |
61 | 57, 60 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑧 ∈ 𝑌 ∧ (𝐹‘𝑧) = 𝑦)) → ¬ ∅ = (◡𝐹 “ 𝑥)) |
62 | 61 | rexlimdvaa 3014 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑥)) → (∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = 𝑦 → ¬ ∅ = (◡𝐹 “ 𝑥))) |
63 | 42, 62 | sylbid 229 |
. . . . . . . . . . . . . 14
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑥)) → (𝑦 ∈ ran 𝐹 → ¬ ∅ = (◡𝐹 “ 𝑥))) |
64 | 63 | con2d 128 |
. . . . . . . . . . . . 13
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ (𝑥 ∈ 𝐿 ∧ 𝑦 ∈ 𝑥)) → (∅ = (◡𝐹 “ 𝑥) → ¬ 𝑦 ∈ ran 𝐹)) |
65 | 64 | expr 641 |
. . . . . . . . . . . 12
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) → (𝑦 ∈ 𝑥 → (∅ = (◡𝐹 “ 𝑥) → ¬ 𝑦 ∈ ran 𝐹))) |
66 | 65 | com23 84 |
. . . . . . . . . . 11
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) → (∅ = (◡𝐹 “ 𝑥) → (𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ ran 𝐹))) |
67 | 66 | impr 647 |
. . . . . . . . . 10
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ (𝑥 ∈ 𝐿 ∧ ∅ = (◡𝐹 “ 𝑥))) → (𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ ran 𝐹)) |
68 | 67 | alrimiv 1842 |
. . . . . . . . 9
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ (𝑥 ∈ 𝐿 ∧ ∅ = (◡𝐹 “ 𝑥))) → ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ ran 𝐹)) |
69 | | imnan 437 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ ran 𝐹) ↔ ¬ (𝑦 ∈ 𝑥 ∧ 𝑦 ∈ ran 𝐹)) |
70 | | elin 3758 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ (𝑥 ∩ ran 𝐹) ↔ (𝑦 ∈ 𝑥 ∧ 𝑦 ∈ ran 𝐹)) |
71 | 69, 70 | xchbinxr 324 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ ran 𝐹) ↔ ¬ 𝑦 ∈ (𝑥 ∩ ran 𝐹)) |
72 | 71 | albii 1737 |
. . . . . . . . . 10
⊢
(∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ ran 𝐹) ↔ ∀𝑦 ¬ 𝑦 ∈ (𝑥 ∩ ran 𝐹)) |
73 | | eq0 3888 |
. . . . . . . . . 10
⊢ ((𝑥 ∩ ran 𝐹) = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ (𝑥 ∩ ran 𝐹)) |
74 | | eqcom 2617 |
. . . . . . . . . 10
⊢ ((𝑥 ∩ ran 𝐹) = ∅ ↔ ∅ = (𝑥 ∩ ran 𝐹)) |
75 | 72, 73, 74 | 3bitr2i 287 |
. . . . . . . . 9
⊢
(∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ ran 𝐹) ↔ ∅ = (𝑥 ∩ ran 𝐹)) |
76 | 68, 75 | sylib 207 |
. . . . . . . 8
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ (𝑥 ∈ 𝐿 ∧ ∅ = (◡𝐹 “ 𝑥))) → ∅ = (𝑥 ∩ ran 𝐹)) |
77 | | simpll2 1094 |
. . . . . . . . 9
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ (𝑥 ∈ 𝐿 ∧ ∅ = (◡𝐹 “ 𝑥))) → 𝐿 ∈ (Fil‘𝑋)) |
78 | | simprl 790 |
. . . . . . . . 9
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ (𝑥 ∈ 𝐿 ∧ ∅ = (◡𝐹 “ 𝑥))) → 𝑥 ∈ 𝐿) |
79 | | simplr 788 |
. . . . . . . . 9
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ (𝑥 ∈ 𝐿 ∧ ∅ = (◡𝐹 “ 𝑥))) → ran 𝐹 ∈ 𝐿) |
80 | | filin 21468 |
. . . . . . . . 9
⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐿 ∧ ran 𝐹 ∈ 𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿) |
81 | 77, 78, 79, 80 | syl3anc 1318 |
. . . . . . . 8
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ (𝑥 ∈ 𝐿 ∧ ∅ = (◡𝐹 “ 𝑥))) → (𝑥 ∩ ran 𝐹) ∈ 𝐿) |
82 | 76, 81 | eqeltrd 2688 |
. . . . . . 7
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ (𝑥 ∈ 𝐿 ∧ ∅ = (◡𝐹 “ 𝑥))) → ∅ ∈ 𝐿) |
83 | 82 | rexlimdvaa 3014 |
. . . . . 6
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (∃𝑥 ∈ 𝐿 ∅ = (◡𝐹 “ 𝑥) → ∅ ∈ 𝐿)) |
84 | 37, 83 | syl5bi 231 |
. . . . 5
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (∅ ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) → ∅ ∈ 𝐿)) |
85 | 34, 84 | mtod 188 |
. . . 4
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ¬ ∅ ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) |
86 | | df-nel 2783 |
. . . 4
⊢ (∅
∉ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ¬ ∅ ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) |
87 | 85, 86 | sylibr 223 |
. . 3
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ∅ ∉ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) |
88 | | vex 3176 |
. . . . . . . . 9
⊢ 𝑟 ∈ V |
89 | 11 | elrnmpt 5293 |
. . . . . . . . 9
⊢ (𝑟 ∈ V → (𝑟 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑟 = (◡𝐹 “ 𝑥))) |
90 | 88, 89 | ax-mp 5 |
. . . . . . . 8
⊢ (𝑟 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑟 = (◡𝐹 “ 𝑥)) |
91 | | imaeq2 5381 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑢 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑢)) |
92 | 91 | eqeq2d 2620 |
. . . . . . . . 9
⊢ (𝑥 = 𝑢 → (𝑟 = (◡𝐹 “ 𝑥) ↔ 𝑟 = (◡𝐹 “ 𝑢))) |
93 | 92 | cbvrexv 3148 |
. . . . . . . 8
⊢
(∃𝑥 ∈
𝐿 𝑟 = (◡𝐹 “ 𝑥) ↔ ∃𝑢 ∈ 𝐿 𝑟 = (◡𝐹 “ 𝑢)) |
94 | 90, 93 | bitri 263 |
. . . . . . 7
⊢ (𝑟 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑢 ∈ 𝐿 𝑟 = (◡𝐹 “ 𝑢)) |
95 | | vex 3176 |
. . . . . . . . 9
⊢ 𝑠 ∈ V |
96 | 11 | elrnmpt 5293 |
. . . . . . . . 9
⊢ (𝑠 ∈ V → (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥))) |
97 | 95, 96 | ax-mp 5 |
. . . . . . . 8
⊢ (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥)) |
98 | | imaeq2 5381 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑣 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑣)) |
99 | 98 | eqeq2d 2620 |
. . . . . . . . 9
⊢ (𝑥 = 𝑣 → (𝑠 = (◡𝐹 “ 𝑥) ↔ 𝑠 = (◡𝐹 “ 𝑣))) |
100 | 99 | cbvrexv 3148 |
. . . . . . . 8
⊢
(∃𝑥 ∈
𝐿 𝑠 = (◡𝐹 “ 𝑥) ↔ ∃𝑣 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑣)) |
101 | 97, 100 | bitri 263 |
. . . . . . 7
⊢ (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑣 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑣)) |
102 | 94, 101 | anbi12i 729 |
. . . . . 6
⊢ ((𝑟 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∧ 𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) ↔ (∃𝑢 ∈ 𝐿 𝑟 = (◡𝐹 “ 𝑢) ∧ ∃𝑣 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑣))) |
103 | | reeanv 3086 |
. . . . . 6
⊢
(∃𝑢 ∈
𝐿 ∃𝑣 ∈ 𝐿 (𝑟 = (◡𝐹 “ 𝑢) ∧ 𝑠 = (◡𝐹 “ 𝑣)) ↔ (∃𝑢 ∈ 𝐿 𝑟 = (◡𝐹 “ 𝑢) ∧ ∃𝑣 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑣))) |
104 | 102, 103 | bitr4i 266 |
. . . . 5
⊢ ((𝑟 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∧ 𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) ↔ ∃𝑢 ∈ 𝐿 ∃𝑣 ∈ 𝐿 (𝑟 = (◡𝐹 “ 𝑢) ∧ 𝑠 = (◡𝐹 “ 𝑣))) |
105 | | filin 21468 |
. . . . . . . . . . . . . 14
⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿) → (𝑢 ∩ 𝑣) ∈ 𝐿) |
106 | 105 | 3expb 1258 |
. . . . . . . . . . . . 13
⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ (𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿)) → (𝑢 ∩ 𝑣) ∈ 𝐿) |
107 | 106 | adantlr 747 |
. . . . . . . . . . . 12
⊢ (((𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿)) → (𝑢 ∩ 𝑣) ∈ 𝐿) |
108 | | eqidd 2611 |
. . . . . . . . . . . 12
⊢ (((𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿)) → (◡𝐹 “ (𝑢 ∩ 𝑣)) = (◡𝐹 “ (𝑢 ∩ 𝑣))) |
109 | | imaeq2 5381 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = (𝑢 ∩ 𝑣) → (◡𝐹 “ 𝑥) = (◡𝐹 “ (𝑢 ∩ 𝑣))) |
110 | 109 | eqeq2d 2620 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝑢 ∩ 𝑣) → ((◡𝐹 “ (𝑢 ∩ 𝑣)) = (◡𝐹 “ 𝑥) ↔ (◡𝐹 “ (𝑢 ∩ 𝑣)) = (◡𝐹 “ (𝑢 ∩ 𝑣)))) |
111 | 110 | rspcev 3282 |
. . . . . . . . . . . 12
⊢ (((𝑢 ∩ 𝑣) ∈ 𝐿 ∧ (◡𝐹 “ (𝑢 ∩ 𝑣)) = (◡𝐹 “ (𝑢 ∩ 𝑣))) → ∃𝑥 ∈ 𝐿 (◡𝐹 “ (𝑢 ∩ 𝑣)) = (◡𝐹 “ 𝑥)) |
112 | 107, 108,
111 | syl2anc 691 |
. . . . . . . . . . 11
⊢ (((𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿)) → ∃𝑥 ∈ 𝐿 (◡𝐹 “ (𝑢 ∩ 𝑣)) = (◡𝐹 “ 𝑥)) |
113 | 112 | 3adantl1 1210 |
. . . . . . . . . 10
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿)) → ∃𝑥 ∈ 𝐿 (◡𝐹 “ (𝑢 ∩ 𝑣)) = (◡𝐹 “ 𝑥)) |
114 | 113 | ad2ant2r 779 |
. . . . . . . . 9
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ ((𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿) ∧ (𝑟 = (◡𝐹 “ 𝑢) ∧ 𝑠 = (◡𝐹 “ 𝑣)))) → ∃𝑥 ∈ 𝐿 (◡𝐹 “ (𝑢 ∩ 𝑣)) = (◡𝐹 “ 𝑥)) |
115 | | simpll1 1093 |
. . . . . . . . . . 11
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ ((𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿) ∧ (𝑟 = (◡𝐹 “ 𝑢) ∧ 𝑠 = (◡𝐹 “ 𝑣)))) → 𝑌 ∈ 𝐴) |
116 | | cnvimass 5404 |
. . . . . . . . . . . . . 14
⊢ (◡𝐹 “ (𝑢 ∩ 𝑣)) ⊆ dom 𝐹 |
117 | 116, 3 | syl5sseq 3616 |
. . . . . . . . . . . . 13
⊢ (𝐹:𝑌⟶𝑋 → (◡𝐹 “ (𝑢 ∩ 𝑣)) ⊆ 𝑌) |
118 | 117 | 3ad2ant3 1077 |
. . . . . . . . . . . 12
⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (◡𝐹 “ (𝑢 ∩ 𝑣)) ⊆ 𝑌) |
119 | 118 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ ((𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿) ∧ (𝑟 = (◡𝐹 “ 𝑢) ∧ 𝑠 = (◡𝐹 “ 𝑣)))) → (◡𝐹 “ (𝑢 ∩ 𝑣)) ⊆ 𝑌) |
120 | 115, 119 | ssexd 4733 |
. . . . . . . . . 10
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ ((𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿) ∧ (𝑟 = (◡𝐹 “ 𝑢) ∧ 𝑠 = (◡𝐹 “ 𝑣)))) → (◡𝐹 “ (𝑢 ∩ 𝑣)) ∈ V) |
121 | 11 | elrnmpt 5293 |
. . . . . . . . . 10
⊢ ((◡𝐹 “ (𝑢 ∩ 𝑣)) ∈ V → ((◡𝐹 “ (𝑢 ∩ 𝑣)) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 (◡𝐹 “ (𝑢 ∩ 𝑣)) = (◡𝐹 “ 𝑥))) |
122 | 120, 121 | syl 17 |
. . . . . . . . 9
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ ((𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿) ∧ (𝑟 = (◡𝐹 “ 𝑢) ∧ 𝑠 = (◡𝐹 “ 𝑣)))) → ((◡𝐹 “ (𝑢 ∩ 𝑣)) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 (◡𝐹 “ (𝑢 ∩ 𝑣)) = (◡𝐹 “ 𝑥))) |
123 | 114, 122 | mpbird 246 |
. . . . . . . 8
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ ((𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿) ∧ (𝑟 = (◡𝐹 “ 𝑢) ∧ 𝑠 = (◡𝐹 “ 𝑣)))) → (◡𝐹 “ (𝑢 ∩ 𝑣)) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) |
124 | | simprrl 800 |
. . . . . . . . . . 11
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ ((𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿) ∧ (𝑟 = (◡𝐹 “ 𝑢) ∧ 𝑠 = (◡𝐹 “ 𝑣)))) → 𝑟 = (◡𝐹 “ 𝑢)) |
125 | | simprrr 801 |
. . . . . . . . . . 11
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ ((𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿) ∧ (𝑟 = (◡𝐹 “ 𝑢) ∧ 𝑠 = (◡𝐹 “ 𝑣)))) → 𝑠 = (◡𝐹 “ 𝑣)) |
126 | 124, 125 | ineq12d 3777 |
. . . . . . . . . 10
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ ((𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿) ∧ (𝑟 = (◡𝐹 “ 𝑢) ∧ 𝑠 = (◡𝐹 “ 𝑣)))) → (𝑟 ∩ 𝑠) = ((◡𝐹 “ 𝑢) ∩ (◡𝐹 “ 𝑣))) |
127 | | funcnvcnv 5870 |
. . . . . . . . . . . . 13
⊢ (Fun
𝐹 → Fun ◡◡𝐹) |
128 | | imain 5888 |
. . . . . . . . . . . . 13
⊢ (Fun
◡◡𝐹 → (◡𝐹 “ (𝑢 ∩ 𝑣)) = ((◡𝐹 “ 𝑢) ∩ (◡𝐹 “ 𝑣))) |
129 | 47, 127, 128 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝐹:𝑌⟶𝑋 → (◡𝐹 “ (𝑢 ∩ 𝑣)) = ((◡𝐹 “ 𝑢) ∩ (◡𝐹 “ 𝑣))) |
130 | 129 | 3ad2ant3 1077 |
. . . . . . . . . . 11
⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (◡𝐹 “ (𝑢 ∩ 𝑣)) = ((◡𝐹 “ 𝑢) ∩ (◡𝐹 “ 𝑣))) |
131 | 130 | ad2antrr 758 |
. . . . . . . . . 10
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ ((𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿) ∧ (𝑟 = (◡𝐹 “ 𝑢) ∧ 𝑠 = (◡𝐹 “ 𝑣)))) → (◡𝐹 “ (𝑢 ∩ 𝑣)) = ((◡𝐹 “ 𝑢) ∩ (◡𝐹 “ 𝑣))) |
132 | 126, 131 | eqtr4d 2647 |
. . . . . . . . 9
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ ((𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿) ∧ (𝑟 = (◡𝐹 “ 𝑢) ∧ 𝑠 = (◡𝐹 “ 𝑣)))) → (𝑟 ∩ 𝑠) = (◡𝐹 “ (𝑢 ∩ 𝑣))) |
133 | | eqimss2 3621 |
. . . . . . . . 9
⊢ ((𝑟 ∩ 𝑠) = (◡𝐹 “ (𝑢 ∩ 𝑣)) → (◡𝐹 “ (𝑢 ∩ 𝑣)) ⊆ (𝑟 ∩ 𝑠)) |
134 | 132, 133 | syl 17 |
. . . . . . . 8
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ ((𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿) ∧ (𝑟 = (◡𝐹 “ 𝑢) ∧ 𝑠 = (◡𝐹 “ 𝑣)))) → (◡𝐹 “ (𝑢 ∩ 𝑣)) ⊆ (𝑟 ∩ 𝑠)) |
135 | | sseq1 3589 |
. . . . . . . . 9
⊢ (𝑡 = (◡𝐹 “ (𝑢 ∩ 𝑣)) → (𝑡 ⊆ (𝑟 ∩ 𝑠) ↔ (◡𝐹 “ (𝑢 ∩ 𝑣)) ⊆ (𝑟 ∩ 𝑠))) |
136 | 135 | rspcev 3282 |
. . . . . . . 8
⊢ (((◡𝐹 “ (𝑢 ∩ 𝑣)) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∧ (◡𝐹 “ (𝑢 ∩ 𝑣)) ⊆ (𝑟 ∩ 𝑠)) → ∃𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))𝑡 ⊆ (𝑟 ∩ 𝑠)) |
137 | 123, 134,
136 | syl2anc 691 |
. . . . . . 7
⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ ((𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿) ∧ (𝑟 = (◡𝐹 “ 𝑢) ∧ 𝑠 = (◡𝐹 “ 𝑣)))) → ∃𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))𝑡 ⊆ (𝑟 ∩ 𝑠)) |
138 | 137 | exp32 629 |
. . . . . 6
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ((𝑢 ∈ 𝐿 ∧ 𝑣 ∈ 𝐿) → ((𝑟 = (◡𝐹 “ 𝑢) ∧ 𝑠 = (◡𝐹 “ 𝑣)) → ∃𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))𝑡 ⊆ (𝑟 ∩ 𝑠)))) |
139 | 138 | rexlimdvv 3019 |
. . . . 5
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (∃𝑢 ∈ 𝐿 ∃𝑣 ∈ 𝐿 (𝑟 = (◡𝐹 “ 𝑢) ∧ 𝑠 = (◡𝐹 “ 𝑣)) → ∃𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))𝑡 ⊆ (𝑟 ∩ 𝑠))) |
140 | 104, 139 | syl5bi 231 |
. . . 4
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ((𝑟 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∧ 𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) → ∃𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))𝑡 ⊆ (𝑟 ∩ 𝑠))) |
141 | 140 | ralrimivv 2953 |
. . 3
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ∀𝑟 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))∀𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))∃𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))𝑡 ⊆ (𝑟 ∩ 𝑠)) |
142 | 31, 87, 141 | 3jca 1235 |
. 2
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ≠ ∅ ∧ ∅ ∉ ran
(𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∧ ∀𝑟 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))∀𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))∃𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))𝑡 ⊆ (𝑟 ∩ 𝑠))) |
143 | | isfbas2 21449 |
. . 3
⊢ (𝑌 ∈ 𝐴 → (ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ (fBas‘𝑌) ↔ (ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ⊆ 𝒫 𝑌 ∧ (ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ≠ ∅ ∧ ∅ ∉ ran
(𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∧ ∀𝑟 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))∀𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))∃𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))𝑡 ⊆ (𝑟 ∩ 𝑠))))) |
144 | 6, 143 | syl 17 |
. 2
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ (fBas‘𝑌) ↔ (ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ⊆ 𝒫 𝑌 ∧ (ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ≠ ∅ ∧ ∅ ∉ ran
(𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∧ ∀𝑟 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))∀𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))∃𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))𝑡 ⊆ (𝑟 ∩ 𝑠))))) |
145 | 14, 142, 144 | mpbir2and 959 |
1
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ (fBas‘𝑌)) |