Step | Hyp | Ref
| Expression |
1 | | df-ov 6817 |
. . . . . . . . . 10
⊢ (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘〈𝑥, 𝑦〉) |
2 | | simprl 811 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ∈ 𝑋) |
3 | | simprr 813 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑦 ∈ 𝑌) |
4 | | cnmpt21.j |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
5 | | cnmpt21.k |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) |
6 | | txtopon 21616 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) |
7 | 4, 5, 6 | syl2anc 696 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) |
8 | | cnmpt21.l |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) |
9 | | cnmpt21.a |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) |
10 | | cnf2 21275 |
. . . . . . . . . . . . . . 15
⊢ (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍) |
11 | 7, 8, 9, 10 | syl3anc 1477 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍) |
12 | | eqid 2760 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) |
13 | 12 | fmpt2 7406 |
. . . . . . . . . . . . . 14
⊢
(∀𝑥 ∈
𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍) |
14 | 11, 13 | sylibr 224 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍) |
15 | | rsp2 3074 |
. . . . . . . . . . . . 13
⊢
(∀𝑥 ∈
𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍)) |
16 | 14, 15 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍)) |
17 | 16 | imp 444 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝐴 ∈ 𝑍) |
18 | 12 | ovmpt4g 6949 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐴 ∈ 𝑍) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴) |
19 | 2, 3, 17, 18 | syl3anc 1477 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴) |
20 | 1, 19 | syl5eqr 2808 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘〈𝑥, 𝑦〉) = 𝐴) |
21 | 20 | fveq2d 6357 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧 ∈ 𝑍 ↦ 𝐵)‘((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘〈𝑥, 𝑦〉)) = ((𝑧 ∈ 𝑍 ↦ 𝐵)‘𝐴)) |
22 | | cnmpt21.c |
. . . . . . . . . . 11
⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶) |
23 | 22 | eleq1d 2824 |
. . . . . . . . . 10
⊢ (𝑧 = 𝐴 → (𝐵 ∈ ∪ 𝑀 ↔ 𝐶 ∈ ∪ 𝑀)) |
24 | | cnmpt21.b |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀)) |
25 | | cntop2 21267 |
. . . . . . . . . . . . . . 15
⊢ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀) → 𝑀 ∈ Top) |
26 | 24, 25 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 ∈ Top) |
27 | | eqid 2760 |
. . . . . . . . . . . . . . 15
⊢ ∪ 𝑀 =
∪ 𝑀 |
28 | 27 | toptopon 20944 |
. . . . . . . . . . . . . 14
⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀)) |
29 | 26, 28 | sylib 208 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀)) |
30 | | cnf2 21275 |
. . . . . . . . . . . . 13
⊢ ((𝐿 ∈ (TopOn‘𝑍) ∧ 𝑀 ∈ (TopOn‘∪ 𝑀)
∧ (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀)) → (𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀) |
31 | 8, 29, 24, 30 | syl3anc 1477 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀) |
32 | | eqid 2760 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ 𝑍 ↦ 𝐵) = (𝑧 ∈ 𝑍 ↦ 𝐵) |
33 | 32 | fmpt 6545 |
. . . . . . . . . . . 12
⊢
(∀𝑧 ∈
𝑍 𝐵 ∈ ∪ 𝑀 ↔ (𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀) |
34 | 31, 33 | sylibr 224 |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑧 ∈ 𝑍 𝐵 ∈ ∪ 𝑀) |
35 | 34 | adantr 472 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ∀𝑧 ∈ 𝑍 𝐵 ∈ ∪ 𝑀) |
36 | 23, 35, 17 | rspcdva 3455 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝐶 ∈ ∪ 𝑀) |
37 | 22, 32 | fvmptg 6443 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑍 ∧ 𝐶 ∈ ∪ 𝑀) → ((𝑧 ∈ 𝑍 ↦ 𝐵)‘𝐴) = 𝐶) |
38 | 17, 36, 37 | syl2anc 696 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧 ∈ 𝑍 ↦ 𝐵)‘𝐴) = 𝐶) |
39 | 21, 38 | eqtrd 2794 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧 ∈ 𝑍 ↦ 𝐵)‘((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘〈𝑥, 𝑦〉)) = 𝐶) |
40 | | opelxpi 5305 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 〈𝑥, 𝑦〉 ∈ (𝑋 × 𝑌)) |
41 | | fvco3 6438 |
. . . . . . . 8
⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍 ∧ 〈𝑥, 𝑦〉 ∈ (𝑋 × 𝑌)) → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑦〉) = ((𝑧 ∈ 𝑍 ↦ 𝐵)‘((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘〈𝑥, 𝑦〉))) |
42 | 11, 40, 41 | syl2an 495 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑦〉) = ((𝑧 ∈ 𝑍 ↦ 𝐵)‘((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘〈𝑥, 𝑦〉))) |
43 | | df-ov 6817 |
. . . . . . . 8
⊢ (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)𝑦) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑦〉) |
44 | | eqid 2760 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) |
45 | 44 | ovmpt4g 6949 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐶 ∈ ∪ 𝑀) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)𝑦) = 𝐶) |
46 | 2, 3, 36, 45 | syl3anc 1477 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)𝑦) = 𝐶) |
47 | 43, 46 | syl5eqr 2808 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑦〉) = 𝐶) |
48 | 39, 42, 47 | 3eqtr4d 2804 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑦〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑦〉)) |
49 | 48 | ralrimivva 3109 |
. . . . 5
⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑦〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑦〉)) |
50 | | nfv 1992 |
. . . . . 6
⊢
Ⅎ𝑢∀𝑦 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑦〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑦〉) |
51 | | nfcv 2902 |
. . . . . . 7
⊢
Ⅎ𝑥𝑌 |
52 | | nfcv 2902 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝑧 ∈ 𝑍 ↦ 𝐵) |
53 | | nfmpt21 6888 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) |
54 | 52, 53 | nfco 5443 |
. . . . . . . . 9
⊢
Ⅎ𝑥((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) |
55 | | nfcv 2902 |
. . . . . . . . 9
⊢
Ⅎ𝑥〈𝑢, 𝑣〉 |
56 | 54, 55 | nffv 6360 |
. . . . . . . 8
⊢
Ⅎ𝑥(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑢, 𝑣〉) |
57 | | nfmpt21 6888 |
. . . . . . . . 9
⊢
Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) |
58 | 57, 55 | nffv 6360 |
. . . . . . . 8
⊢
Ⅎ𝑥((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑢, 𝑣〉) |
59 | 56, 58 | nfeq 2914 |
. . . . . . 7
⊢
Ⅎ𝑥(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑢, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑢, 𝑣〉) |
60 | 51, 59 | nfral 3083 |
. . . . . 6
⊢
Ⅎ𝑥∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑢, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑢, 𝑣〉) |
61 | | nfv 1992 |
. . . . . . . 8
⊢
Ⅎ𝑣(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑦〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑦〉) |
62 | | nfcv 2902 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦(𝑧 ∈ 𝑍 ↦ 𝐵) |
63 | | nfmpt22 6889 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) |
64 | 62, 63 | nfco 5443 |
. . . . . . . . . 10
⊢
Ⅎ𝑦((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) |
65 | | nfcv 2902 |
. . . . . . . . . 10
⊢
Ⅎ𝑦〈𝑥, 𝑣〉 |
66 | 64, 65 | nffv 6360 |
. . . . . . . . 9
⊢
Ⅎ𝑦(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑣〉) |
67 | | nfmpt22 6889 |
. . . . . . . . . 10
⊢
Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) |
68 | 67, 65 | nffv 6360 |
. . . . . . . . 9
⊢
Ⅎ𝑦((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑣〉) |
69 | 66, 68 | nfeq 2914 |
. . . . . . . 8
⊢
Ⅎ𝑦(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑣〉) |
70 | | opeq2 4554 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑣 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝑣〉) |
71 | 70 | fveq2d 6357 |
. . . . . . . . 9
⊢ (𝑦 = 𝑣 → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑦〉) = (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑣〉)) |
72 | 70 | fveq2d 6357 |
. . . . . . . . 9
⊢ (𝑦 = 𝑣 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑦〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑣〉)) |
73 | 71, 72 | eqeq12d 2775 |
. . . . . . . 8
⊢ (𝑦 = 𝑣 → ((((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑦〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑦〉) ↔ (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑣〉))) |
74 | 61, 69, 73 | cbvral 3306 |
. . . . . . 7
⊢
(∀𝑦 ∈
𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑦〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑦〉) ↔ ∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑣〉)) |
75 | | opeq1 4553 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑢 → 〈𝑥, 𝑣〉 = 〈𝑢, 𝑣〉) |
76 | 75 | fveq2d 6357 |
. . . . . . . . 9
⊢ (𝑥 = 𝑢 → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑣〉) = (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑢, 𝑣〉)) |
77 | 75 | fveq2d 6357 |
. . . . . . . . 9
⊢ (𝑥 = 𝑢 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑢, 𝑣〉)) |
78 | 76, 77 | eqeq12d 2775 |
. . . . . . . 8
⊢ (𝑥 = 𝑢 → ((((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑣〉) ↔ (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑢, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑢, 𝑣〉))) |
79 | 78 | ralbidv 3124 |
. . . . . . 7
⊢ (𝑥 = 𝑢 → (∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑣〉) ↔ ∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑢, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑢, 𝑣〉))) |
80 | 74, 79 | syl5bb 272 |
. . . . . 6
⊢ (𝑥 = 𝑢 → (∀𝑦 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑦〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑦〉) ↔ ∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑢, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑢, 𝑣〉))) |
81 | 50, 60, 80 | cbvral 3306 |
. . . . 5
⊢
(∀𝑥 ∈
𝑋 ∀𝑦 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑦〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑦〉) ↔ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑢, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑢, 𝑣〉)) |
82 | 49, 81 | sylib 208 |
. . . 4
⊢ (𝜑 → ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑢, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑢, 𝑣〉)) |
83 | | fveq2 6353 |
. . . . . 6
⊢ (𝑤 = 〈𝑢, 𝑣〉 → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑢, 𝑣〉)) |
84 | | fveq2 6353 |
. . . . . 6
⊢ (𝑤 = 〈𝑢, 𝑣〉 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘𝑤) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑢, 𝑣〉)) |
85 | 83, 84 | eqeq12d 2775 |
. . . . 5
⊢ (𝑤 = 〈𝑢, 𝑣〉 → ((((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘𝑤) ↔ (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑢, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑢, 𝑣〉))) |
86 | 85 | ralxp 5419 |
. . . 4
⊢
(∀𝑤 ∈
(𝑋 × 𝑌)(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘𝑤) ↔ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑢, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑢, 𝑣〉)) |
87 | 82, 86 | sylibr 224 |
. . 3
⊢ (𝜑 → ∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘𝑤)) |
88 | | fco 6219 |
. . . . . 6
⊢ (((𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀 ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍) → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)):(𝑋 × 𝑌)⟶∪ 𝑀) |
89 | 31, 11, 88 | syl2anc 696 |
. . . . 5
⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)):(𝑋 × 𝑌)⟶∪ 𝑀) |
90 | | ffn 6206 |
. . . . 5
⊢ (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)):(𝑋 × 𝑌)⟶∪ 𝑀 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) Fn (𝑋 × 𝑌)) |
91 | 89, 90 | syl 17 |
. . . 4
⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) Fn (𝑋 × 𝑌)) |
92 | 36 | ralrimivva 3109 |
. . . . . 6
⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ ∪ 𝑀) |
93 | 44 | fmpt2 7406 |
. . . . . 6
⊢
(∀𝑥 ∈
𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ ∪ 𝑀 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶):(𝑋 × 𝑌)⟶∪ 𝑀) |
94 | 92, 93 | sylib 208 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶):(𝑋 × 𝑌)⟶∪ 𝑀) |
95 | | ffn 6206 |
. . . . 5
⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶):(𝑋 × 𝑌)⟶∪ 𝑀 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) Fn (𝑋 × 𝑌)) |
96 | 94, 95 | syl 17 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) Fn (𝑋 × 𝑌)) |
97 | | eqfnfv 6475 |
. . . 4
⊢ ((((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) Fn (𝑋 × 𝑌) ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) Fn (𝑋 × 𝑌)) → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) ↔ ∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘𝑤))) |
98 | 91, 96, 97 | syl2anc 696 |
. . 3
⊢ (𝜑 → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) ↔ ∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘𝑤))) |
99 | 87, 98 | mpbird 247 |
. 2
⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)) |
100 | | cnco 21292 |
. . 3
⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀)) → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) |
101 | 9, 24, 100 | syl2anc 696 |
. 2
⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) |
102 | 99, 101 | eqeltrrd 2840 |
1
⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) |