Proof of Theorem mapfienlem2
Step | Hyp | Ref
| Expression |
1 | | mapfien.z |
. . . 4
⊢ (𝜑 → 𝑍 ∈ 𝐵) |
2 | 1 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑍 ∈ 𝐵) |
3 | | mapfien.w |
. . . . 5
⊢ 𝑊 = (𝐺‘𝑍) |
4 | | mapfien.g |
. . . . . . 7
⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷) |
5 | | f1of 6050 |
. . . . . . 7
⊢ (𝐺:𝐵–1-1-onto→𝐷 → 𝐺:𝐵⟶𝐷) |
6 | 4, 5 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐺:𝐵⟶𝐷) |
7 | 6, 1 | ffvelrnd 6268 |
. . . . 5
⊢ (𝜑 → (𝐺‘𝑍) ∈ 𝐷) |
8 | 3, 7 | syl5eqel 2692 |
. . . 4
⊢ (𝜑 → 𝑊 ∈ 𝐷) |
9 | 8 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑊 ∈ 𝐷) |
10 | | elrabi 3328 |
. . . . . 6
⊢ (𝑔 ∈ {𝑥 ∈ (𝐷 ↑𝑚 𝐶) ∣ 𝑥 finSupp 𝑊} → 𝑔 ∈ (𝐷 ↑𝑚 𝐶)) |
11 | | elmapi 7765 |
. . . . . 6
⊢ (𝑔 ∈ (𝐷 ↑𝑚 𝐶) → 𝑔:𝐶⟶𝐷) |
12 | 10, 11 | syl 17 |
. . . . 5
⊢ (𝑔 ∈ {𝑥 ∈ (𝐷 ↑𝑚 𝐶) ∣ 𝑥 finSupp 𝑊} → 𝑔:𝐶⟶𝐷) |
13 | | mapfien.t |
. . . . 5
⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑𝑚 𝐶) ∣ 𝑥 finSupp 𝑊} |
14 | 12, 13 | eleq2s 2706 |
. . . 4
⊢ (𝑔 ∈ 𝑇 → 𝑔:𝐶⟶𝐷) |
15 | 14 | adantl 481 |
. . 3
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔:𝐶⟶𝐷) |
16 | | f1ocnv 6062 |
. . . . 5
⊢ (𝐺:𝐵–1-1-onto→𝐷 → ◡𝐺:𝐷–1-1-onto→𝐵) |
17 | | f1of 6050 |
. . . . 5
⊢ (◡𝐺:𝐷–1-1-onto→𝐵 → ◡𝐺:𝐷⟶𝐵) |
18 | 4, 16, 17 | 3syl 18 |
. . . 4
⊢ (𝜑 → ◡𝐺:𝐷⟶𝐵) |
19 | 18 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ◡𝐺:𝐷⟶𝐵) |
20 | | ssid 3587 |
. . . 4
⊢ 𝐷 ⊆ 𝐷 |
21 | 20 | a1i 11 |
. . 3
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝐷 ⊆ 𝐷) |
22 | | mapfien.c |
. . . 4
⊢ (𝜑 → 𝐶 ∈ V) |
23 | 22 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝐶 ∈ V) |
24 | | mapfien.d |
. . . 4
⊢ (𝜑 → 𝐷 ∈ V) |
25 | 24 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝐷 ∈ V) |
26 | | breq1 4586 |
. . . . . . 7
⊢ (𝑥 = 𝑔 → (𝑥 finSupp 𝑊 ↔ 𝑔 finSupp 𝑊)) |
27 | 26 | elrab 3331 |
. . . . . 6
⊢ (𝑔 ∈ {𝑥 ∈ (𝐷 ↑𝑚 𝐶) ∣ 𝑥 finSupp 𝑊} ↔ (𝑔 ∈ (𝐷 ↑𝑚 𝐶) ∧ 𝑔 finSupp 𝑊)) |
28 | 27 | simprbi 479 |
. . . . 5
⊢ (𝑔 ∈ {𝑥 ∈ (𝐷 ↑𝑚 𝐶) ∣ 𝑥 finSupp 𝑊} → 𝑔 finSupp 𝑊) |
29 | 28, 13 | eleq2s 2706 |
. . . 4
⊢ (𝑔 ∈ 𝑇 → 𝑔 finSupp 𝑊) |
30 | 29 | adantl 481 |
. . 3
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔 finSupp 𝑊) |
31 | 4, 1 | jca 553 |
. . . . . 6
⊢ (𝜑 → (𝐺:𝐵–1-1-onto→𝐷 ∧ 𝑍 ∈ 𝐵)) |
32 | 3 | eqcomi 2619 |
. . . . . . 7
⊢ (𝐺‘𝑍) = 𝑊 |
33 | 32 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (𝐺‘𝑍) = 𝑊) |
34 | 31, 33 | jca 553 |
. . . . 5
⊢ (𝜑 → ((𝐺:𝐵–1-1-onto→𝐷 ∧ 𝑍 ∈ 𝐵) ∧ (𝐺‘𝑍) = 𝑊)) |
35 | 34 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((𝐺:𝐵–1-1-onto→𝐷 ∧ 𝑍 ∈ 𝐵) ∧ (𝐺‘𝑍) = 𝑊)) |
36 | | f1ocnvfv 6434 |
. . . . 5
⊢ ((𝐺:𝐵–1-1-onto→𝐷 ∧ 𝑍 ∈ 𝐵) → ((𝐺‘𝑍) = 𝑊 → (◡𝐺‘𝑊) = 𝑍)) |
37 | 36 | imp 444 |
. . . 4
⊢ (((𝐺:𝐵–1-1-onto→𝐷 ∧ 𝑍 ∈ 𝐵) ∧ (𝐺‘𝑍) = 𝑊) → (◡𝐺‘𝑊) = 𝑍) |
38 | 35, 37 | syl 17 |
. . 3
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (◡𝐺‘𝑊) = 𝑍) |
39 | 2, 9, 15, 19, 21, 23, 25, 30, 38 | fsuppcor 8192 |
. 2
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (◡𝐺 ∘ 𝑔) finSupp 𝑍) |
40 | | mapfien.f |
. . . 4
⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) |
41 | | f1ocnv 6062 |
. . . 4
⊢ (𝐹:𝐶–1-1-onto→𝐴 → ◡𝐹:𝐴–1-1-onto→𝐶) |
42 | | f1of1 6049 |
. . . 4
⊢ (◡𝐹:𝐴–1-1-onto→𝐶 → ◡𝐹:𝐴–1-1→𝐶) |
43 | 40, 41, 42 | 3syl 18 |
. . 3
⊢ (𝜑 → ◡𝐹:𝐴–1-1→𝐶) |
44 | 43 | adantr 480 |
. 2
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ◡𝐹:𝐴–1-1→𝐶) |
45 | | mapfien.b |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ V) |
46 | 6, 45 | jca 553 |
. . . 4
⊢ (𝜑 → (𝐺:𝐵⟶𝐷 ∧ 𝐵 ∈ V)) |
47 | | fex 6394 |
. . . 4
⊢ ((𝐺:𝐵⟶𝐷 ∧ 𝐵 ∈ V) → 𝐺 ∈ V) |
48 | | cnvexg 7005 |
. . . 4
⊢ (𝐺 ∈ V → ◡𝐺 ∈ V) |
49 | 46, 47, 48 | 3syl 18 |
. . 3
⊢ (𝜑 → ◡𝐺 ∈ V) |
50 | | coexg 7010 |
. . 3
⊢ ((◡𝐺 ∈ V ∧ 𝑔 ∈ 𝑇) → (◡𝐺 ∘ 𝑔) ∈ V) |
51 | 49, 50 | sylan 487 |
. 2
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (◡𝐺 ∘ 𝑔) ∈ V) |
52 | 39, 44, 2, 51 | fsuppco 8190 |
1
⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) finSupp 𝑍) |