Step | Hyp | Ref
| Expression |
1 | | ovexd 7310 |
. 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ↑m 𝐵) ↑m 𝐶) ∈ V) |
2 | | ovexd 7310 |
. 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 ↑m (𝐵 × 𝐶)) ∈ V) |
3 | | elmapi 8637 |
. . . . . . . . . 10
⊢ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) → 𝑓:𝐶⟶(𝐴 ↑m 𝐵)) |
4 | 3 | ffvelrnda 6961 |
. . . . . . . . 9
⊢ ((𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ 𝐶) → (𝑓‘𝑦) ∈ (𝐴 ↑m 𝐵)) |
5 | | elmapi 8637 |
. . . . . . . . 9
⊢ ((𝑓‘𝑦) ∈ (𝐴 ↑m 𝐵) → (𝑓‘𝑦):𝐵⟶𝐴) |
6 | 4, 5 | syl 17 |
. . . . . . . 8
⊢ ((𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ 𝐶) → (𝑓‘𝑦):𝐵⟶𝐴) |
7 | 6 | ffvelrnda 6961 |
. . . . . . 7
⊢ (((𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝐵) → ((𝑓‘𝑦)‘𝑥) ∈ 𝐴) |
8 | 7 | an32s 649 |
. . . . . 6
⊢ (((𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶) → ((𝑓‘𝑦)‘𝑥) ∈ 𝐴) |
9 | 8 | ralrimiva 3103 |
. . . . 5
⊢ ((𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) ∈ 𝐴) |
10 | 9 | ralrimiva 3103 |
. . . 4
⊢ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) ∈ 𝐴) |
11 | | eqid 2738 |
. . . . 5
⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) |
12 | 11 | fmpo 7908 |
. . . 4
⊢
(∀𝑥 ∈
𝐵 ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) ∈ 𝐴 ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴) |
13 | 10, 12 | sylib 217 |
. . 3
⊢ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴) |
14 | | simp1 1135 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐴 ∈ 𝑉) |
15 | | xpexg 7600 |
. . . . 5
⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐵 × 𝐶) ∈ V) |
16 | 15 | 3adant1 1129 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐵 × 𝐶) ∈ V) |
17 | | elmapg 8628 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 × 𝐶) ∈ V) → ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) ∈ (𝐴 ↑m (𝐵 × 𝐶)) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴)) |
18 | 14, 16, 17 | syl2anc 584 |
. . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) ∈ (𝐴 ↑m (𝐵 × 𝐶)) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)):(𝐵 × 𝐶)⟶𝐴)) |
19 | 13, 18 | syl5ibr 245 |
. 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) ∈ (𝐴 ↑m (𝐵 × 𝐶)))) |
20 | | elmapi 8637 |
. . . . . . . . 9
⊢ (𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)) → 𝑔:(𝐵 × 𝐶)⟶𝐴) |
21 | 20 | adantl 482 |
. . . . . . . 8
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶))) → 𝑔:(𝐵 × 𝐶)⟶𝐴) |
22 | | fovrn 7442 |
. . . . . . . . . 10
⊢ ((𝑔:(𝐵 × 𝐶)⟶𝐴 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝑥𝑔𝑦) ∈ 𝐴) |
23 | 22 | 3expa 1117 |
. . . . . . . . 9
⊢ (((𝑔:(𝐵 × 𝐶)⟶𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶) → (𝑥𝑔𝑦) ∈ 𝐴) |
24 | 23 | an32s 649 |
. . . . . . . 8
⊢ (((𝑔:(𝐵 × 𝐶)⟶𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝐵) → (𝑥𝑔𝑦) ∈ 𝐴) |
25 | 21, 24 | sylanl1 677 |
. . . . . . 7
⊢
(((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶))) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 ∈ 𝐵) → (𝑥𝑔𝑦) ∈ 𝐴) |
26 | 25 | fmpttd 6989 |
. . . . . 6
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶))) ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)):𝐵⟶𝐴) |
27 | | elmapg 8628 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴 ↑m 𝐵) ↔ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)):𝐵⟶𝐴)) |
28 | 27 | 3adant3 1131 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴 ↑m 𝐵) ↔ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)):𝐵⟶𝐴)) |
29 | 28 | ad2antrr 723 |
. . . . . 6
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶))) ∧ 𝑦 ∈ 𝐶) → ((𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴 ↑m 𝐵) ↔ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)):𝐵⟶𝐴)) |
30 | 26, 29 | mpbird 256 |
. . . . 5
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶))) ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) ∈ (𝐴 ↑m 𝐵)) |
31 | 30 | fmpttd 6989 |
. . . 4
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶))) → (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴 ↑m 𝐵)) |
32 | 31 | ex 413 |
. . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)) → (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴 ↑m 𝐵))) |
33 | | ovex 7308 |
. . . 4
⊢ (𝐴 ↑m 𝐵) ∈ V |
34 | | simp3 1137 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ 𝑋) |
35 | | elmapg 8628 |
. . . 4
⊢ (((𝐴 ↑m 𝐵) ∈ V ∧ 𝐶 ∈ 𝑋) → ((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ↔ (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴 ↑m 𝐵))) |
36 | 33, 34, 35 | sylancr 587 |
. . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ↔ (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))):𝐶⟶(𝐴 ↑m 𝐵))) |
37 | 32, 36 | sylibrd 258 |
. 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)) → (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶))) |
38 | | elmapfn 8653 |
. . . . . . . 8
⊢ (𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)) → 𝑔 Fn (𝐵 × 𝐶)) |
39 | 38 | ad2antll 726 |
. . . . . . 7
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) → 𝑔 Fn (𝐵 × 𝐶)) |
40 | | fnov 7405 |
. . . . . . 7
⊢ (𝑔 Fn (𝐵 × 𝐶) ↔ 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑔𝑦))) |
41 | 39, 40 | sylib 217 |
. . . . . 6
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) → 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑔𝑦))) |
42 | | simp3 1137 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ 𝐶) |
43 | 26 | adantlrl 717 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)):𝐵⟶𝐴) |
44 | 43 | 3adant2 1130 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)):𝐵⟶𝐴) |
45 | | simp1l2 1266 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝐵 ∈ 𝑊) |
46 | | simp1l1 1265 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉) |
47 | | fex2 7780 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)):𝐵⟶𝐴 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) ∈ V) |
48 | 44, 45, 46, 47 | syl3anc 1370 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) ∈ V) |
49 | | eqid 2738 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) |
50 | 49 | fvmpt2 6886 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) ∈ V) → ((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦) = (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) |
51 | 42, 48, 50 | syl2anc 584 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → ((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦) = (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) |
52 | 51 | fveq1d 6776 |
. . . . . . . 8
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥) = ((𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))‘𝑥)) |
53 | | simp2 1136 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝑥 ∈ 𝐵) |
54 | | ovex 7308 |
. . . . . . . . 9
⊢ (𝑥𝑔𝑦) ∈ V |
55 | | eqid 2738 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) |
56 | 55 | fvmpt2 6886 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝐵 ∧ (𝑥𝑔𝑦) ∈ V) → ((𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))‘𝑥) = (𝑥𝑔𝑦)) |
57 | 53, 54, 56 | sylancl 586 |
. . . . . . . 8
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → ((𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))‘𝑥) = (𝑥𝑔𝑦)) |
58 | 52, 57 | eqtrd 2778 |
. . . . . . 7
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥) = (𝑥𝑔𝑦)) |
59 | 58 | mpoeq3dva 7352 |
. . . . . 6
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑔𝑦))) |
60 | 41, 59 | eqtr4d 2781 |
. . . . 5
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) → 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))) |
61 | | eqid 2738 |
. . . . . . 7
⊢ 𝐵 = 𝐵 |
62 | | nfcv 2907 |
. . . . . . . . . 10
⊢
Ⅎ𝑥𝐶 |
63 | | nfmpt1 5182 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) |
64 | 62, 63 | nfmpt 5181 |
. . . . . . . . 9
⊢
Ⅎ𝑥(𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) |
65 | 64 | nfeq2 2924 |
. . . . . . . 8
⊢
Ⅎ𝑥 𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) |
66 | | nfmpt1 5182 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑦(𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) |
67 | 66 | nfeq2 2924 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦 𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) |
68 | | fveq1 6773 |
. . . . . . . . . . . . 13
⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → (𝑓‘𝑦) = ((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)) |
69 | 68 | fveq1d 6776 |
. . . . . . . . . . . 12
⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)) |
70 | 69 | a1d 25 |
. . . . . . . . . . 11
⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → (𝑦 ∈ 𝐶 → ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))) |
71 | 67, 70 | ralrimi 3141 |
. . . . . . . . . 10
⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)) |
72 | | eqid 2738 |
. . . . . . . . . 10
⊢ 𝐶 = 𝐶 |
73 | 71, 72 | jctil 520 |
. . . . . . . . 9
⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → (𝐶 = 𝐶 ∧ ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))) |
74 | 73 | a1d 25 |
. . . . . . . 8
⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → (𝑥 ∈ 𝐵 → (𝐶 = 𝐶 ∧ ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))) |
75 | 65, 74 | ralrimi 3141 |
. . . . . . 7
⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → ∀𝑥 ∈ 𝐵 (𝐶 = 𝐶 ∧ ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))) |
76 | | mpoeq123 7347 |
. . . . . . 7
⊢ ((𝐵 = 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝐶 = 𝐶 ∧ ∀𝑦 ∈ 𝐶 ((𝑓‘𝑦)‘𝑥) = (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))) |
77 | 61, 75, 76 | sylancr 587 |
. . . . . 6
⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥))) |
78 | 77 | eqeq2d 2749 |
. . . . 5
⊢ (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) ↔ 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (((𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)))‘𝑦)‘𝑥)))) |
79 | 60, 78 | syl5ibrcom 246 |
. . . 4
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) → (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) → 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)))) |
80 | 3 | ad2antrl 725 |
. . . . . . 7
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) → 𝑓:𝐶⟶(𝐴 ↑m 𝐵)) |
81 | 80 | feqmptd 6837 |
. . . . . 6
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) → 𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑓‘𝑦))) |
82 | | simprl 768 |
. . . . . . . . 9
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) → 𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶)) |
83 | 82, 6 | sylan 580 |
. . . . . . . 8
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) ∧ 𝑦 ∈ 𝐶) → (𝑓‘𝑦):𝐵⟶𝐴) |
84 | 83 | feqmptd 6837 |
. . . . . . 7
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) ∧ 𝑦 ∈ 𝐶) → (𝑓‘𝑦) = (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥))) |
85 | 84 | mpteq2dva 5174 |
. . . . . 6
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) → (𝑦 ∈ 𝐶 ↦ (𝑓‘𝑦)) = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥)))) |
86 | 81, 85 | eqtrd 2778 |
. . . . 5
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) → 𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥)))) |
87 | | nfmpo2 7356 |
. . . . . . . . 9
⊢
Ⅎ𝑦(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) |
88 | 87 | nfeq2 2924 |
. . . . . . . 8
⊢
Ⅎ𝑦 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) |
89 | | eqidd 2739 |
. . . . . . . . 9
⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → 𝐵 = 𝐵) |
90 | | nfmpo1 7355 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) |
91 | 90 | nfeq2 2924 |
. . . . . . . . . 10
⊢
Ⅎ𝑥 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) |
92 | | nfv 1917 |
. . . . . . . . . 10
⊢
Ⅎ𝑥 𝑦 ∈ 𝐶 |
93 | | fvex 6787 |
. . . . . . . . . . . . 13
⊢ ((𝑓‘𝑦)‘𝑥) ∈ V |
94 | 11 | ovmpt4g 7420 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ∧ ((𝑓‘𝑦)‘𝑥) ∈ V) → (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))𝑦) = ((𝑓‘𝑦)‘𝑥)) |
95 | 93, 94 | mp3an3 1449 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))𝑦) = ((𝑓‘𝑦)‘𝑥)) |
96 | | oveq 7281 |
. . . . . . . . . . . . 13
⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → (𝑥𝑔𝑦) = (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))𝑦)) |
97 | 96 | eqeq1d 2740 |
. . . . . . . . . . . 12
⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → ((𝑥𝑔𝑦) = ((𝑓‘𝑦)‘𝑥) ↔ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))𝑦) = ((𝑓‘𝑦)‘𝑥))) |
98 | 95, 97 | syl5ibr 245 |
. . . . . . . . . . 11
⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝑥𝑔𝑦) = ((𝑓‘𝑦)‘𝑥))) |
99 | 98 | expcomd 417 |
. . . . . . . . . 10
⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → (𝑦 ∈ 𝐶 → (𝑥 ∈ 𝐵 → (𝑥𝑔𝑦) = ((𝑓‘𝑦)‘𝑥)))) |
100 | 91, 92, 99 | ralrimd 3143 |
. . . . . . . . 9
⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → (𝑦 ∈ 𝐶 → ∀𝑥 ∈ 𝐵 (𝑥𝑔𝑦) = ((𝑓‘𝑦)‘𝑥))) |
101 | | mpteq12 5166 |
. . . . . . . . 9
⊢ ((𝐵 = 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝑥𝑔𝑦) = ((𝑓‘𝑦)‘𝑥)) → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥))) |
102 | 89, 100, 101 | syl6an 681 |
. . . . . . . 8
⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → (𝑦 ∈ 𝐶 → (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥)))) |
103 | 88, 102 | ralrimi 3141 |
. . . . . . 7
⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → ∀𝑦 ∈ 𝐶 (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥))) |
104 | | mpteq12 5166 |
. . . . . . 7
⊢ ((𝐶 = 𝐶 ∧ ∀𝑦 ∈ 𝐶 (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦)) = (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥))) → (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥)))) |
105 | 72, 103, 104 | sylancr 587 |
. . . . . 6
⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥)))) |
106 | 105 | eqeq2d 2749 |
. . . . 5
⊢ (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) ↔ 𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ ((𝑓‘𝑦)‘𝑥))))) |
107 | 86, 106 | syl5ibrcom 246 |
. . . 4
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) → (𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)) → 𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))))) |
108 | 79, 107 | impbid 211 |
. . 3
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶)))) → (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) ↔ 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥)))) |
109 | 108 | ex 413 |
. 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝑓 ∈ ((𝐴 ↑m 𝐵) ↑m 𝐶) ∧ 𝑔 ∈ (𝐴 ↑m (𝐵 × 𝐶))) → (𝑓 = (𝑦 ∈ 𝐶 ↦ (𝑥 ∈ 𝐵 ↦ (𝑥𝑔𝑦))) ↔ 𝑔 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ ((𝑓‘𝑦)‘𝑥))))) |
110 | 1, 2, 19, 37, 109 | en3d 8777 |
1
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ↑m 𝐵) ↑m 𝐶) ≈ (𝐴 ↑m (𝐵 × 𝐶))) |