Step | Hyp | Ref
| Expression |
1 | | df-ov 5828 |
. . . . . . . . . 10
⊢ (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘〈𝑥, 𝑦〉) |
2 | | simprl 521 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ∈ 𝑋) |
3 | | simprr 522 |
. . . . . . . . . . 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 12704 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) |
7 | 4, 5, 6 | syl2anc 409 |
. . . . . . . . . . . . . . 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 12647 |
. . . . . . . . . . . . . . 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 1220 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍) |
12 | | eqid 2157 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) |
13 | 12 | fmpo 6150 |
. . . . . . . . . . . . . 14
⊢
(∀𝑥 ∈
𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶𝑍) |
14 | 11, 13 | sylibr 133 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍) |
15 | | rsp2 2507 |
. . . . . . . . . . . . 13
⊢
(∀𝑥 ∈
𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ 𝑍 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍)) |
16 | 14, 15 | syl 14 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑍)) |
17 | 16 | imp 123 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝐴 ∈ 𝑍) |
18 | 12 | ovmpt4g 5944 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐴 ∈ 𝑍) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴) |
19 | 2, 3, 17, 18 | syl3anc 1220 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴) |
20 | 1, 19 | eqtr3id 2204 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘〈𝑥, 𝑦〉) = 𝐴) |
21 | 20 | fveq2d 5473 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧 ∈ 𝑍 ↦ 𝐵)‘((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘〈𝑥, 𝑦〉)) = ((𝑧 ∈ 𝑍 ↦ 𝐵)‘𝐴)) |
22 | | eqid 2157 |
. . . . . . . . 9
⊢ (𝑧 ∈ 𝑍 ↦ 𝐵) = (𝑧 ∈ 𝑍 ↦ 𝐵) |
23 | | cnmpt21.c |
. . . . . . . . 9
⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶) |
24 | 23 | eleq1d 2226 |
. . . . . . . . . 10
⊢ (𝑧 = 𝐴 → (𝐵 ∈ ∪ 𝑀 ↔ 𝐶 ∈ ∪ 𝑀)) |
25 | | cnmpt21.b |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀)) |
26 | | cntop2 12644 |
. . . . . . . . . . . . . . 15
⊢ ((𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀) → 𝑀 ∈ Top) |
27 | 25, 26 | syl 14 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 ∈ Top) |
28 | | toptopon2 12459 |
. . . . . . . . . . . . . 14
⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀)) |
29 | 27, 28 | sylib 121 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀)) |
30 | | cnf2 12647 |
. . . . . . . . . . . . 13
⊢ ((𝐿 ∈ (TopOn‘𝑍) ∧ 𝑀 ∈ (TopOn‘∪ 𝑀)
∧ (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀)) → (𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀) |
31 | 8, 29, 25, 30 | syl3anc 1220 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀) |
32 | 22 | fmpt 5618 |
. . . . . . . . . . . 12
⊢
(∀𝑧 ∈
𝑍 𝐵 ∈ ∪ 𝑀 ↔ (𝑧 ∈ 𝑍 ↦ 𝐵):𝑍⟶∪ 𝑀) |
33 | 31, 32 | sylibr 133 |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑧 ∈ 𝑍 𝐵 ∈ ∪ 𝑀) |
34 | 33 | adantr 274 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ∀𝑧 ∈ 𝑍 𝐵 ∈ ∪ 𝑀) |
35 | 24, 34, 17 | rspcdva 2821 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝐶 ∈ ∪ 𝑀) |
36 | 22, 23, 17, 35 | fvmptd3 5562 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧 ∈ 𝑍 ↦ 𝐵)‘𝐴) = 𝐶) |
37 | 21, 36 | eqtrd 2190 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑧 ∈ 𝑍 ↦ 𝐵)‘((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘〈𝑥, 𝑦〉)) = 𝐶) |
38 | | opelxpi 4619 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 〈𝑥, 𝑦〉 ∈ (𝑋 × 𝑌)) |
39 | | fvco3 5540 |
. . . . . . . 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〉))) |
40 | 11, 38, 39 | syl2an 287 |
. . . . . . 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〉))) |
41 | | df-ov 5828 |
. . . . . . . 8
⊢ (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)𝑦) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑦〉) |
42 | | eqid 2157 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) |
43 | 42 | ovmpt4g 5944 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐶 ∈ ∪ 𝑀) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)𝑦) = 𝐶) |
44 | 2, 3, 35, 43 | syl3anc 1220 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)𝑦) = 𝐶) |
45 | 41, 44 | eqtr3id 2204 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑦〉) = 𝐶) |
46 | 37, 40, 45 | 3eqtr4d 2200 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑦〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑦〉)) |
47 | 46 | ralrimivva 2539 |
. . . . 5
⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑦〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑦〉)) |
48 | | nfv 1508 |
. . . . . 6
⊢
Ⅎ𝑢∀𝑦 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑦〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑦〉) |
49 | | nfcv 2299 |
. . . . . . 7
⊢
Ⅎ𝑥𝑌 |
50 | | nfcv 2299 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝑧 ∈ 𝑍 ↦ 𝐵) |
51 | | nfmpo1 5889 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) |
52 | 50, 51 | nfco 4752 |
. . . . . . . . 9
⊢
Ⅎ𝑥((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) |
53 | | nfcv 2299 |
. . . . . . . . 9
⊢
Ⅎ𝑥〈𝑢, 𝑣〉 |
54 | 52, 53 | nffv 5479 |
. . . . . . . 8
⊢
Ⅎ𝑥(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑢, 𝑣〉) |
55 | | nfmpo1 5889 |
. . . . . . . . 9
⊢
Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) |
56 | 55, 53 | nffv 5479 |
. . . . . . . 8
⊢
Ⅎ𝑥((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑢, 𝑣〉) |
57 | 54, 56 | nfeq 2307 |
. . . . . . 7
⊢
Ⅎ𝑥(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑢, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑢, 𝑣〉) |
58 | 49, 57 | nfralxy 2495 |
. . . . . 6
⊢
Ⅎ𝑥∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑢, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑢, 𝑣〉) |
59 | | nfv 1508 |
. . . . . . . 8
⊢
Ⅎ𝑣(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑦〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑦〉) |
60 | | nfcv 2299 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦(𝑧 ∈ 𝑍 ↦ 𝐵) |
61 | | nfmpo2 5890 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) |
62 | 60, 61 | nfco 4752 |
. . . . . . . . . 10
⊢
Ⅎ𝑦((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) |
63 | | nfcv 2299 |
. . . . . . . . . 10
⊢
Ⅎ𝑦〈𝑥, 𝑣〉 |
64 | 62, 63 | nffv 5479 |
. . . . . . . . 9
⊢
Ⅎ𝑦(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑣〉) |
65 | | nfmpo2 5890 |
. . . . . . . . . 10
⊢
Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) |
66 | 65, 63 | nffv 5479 |
. . . . . . . . 9
⊢
Ⅎ𝑦((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑣〉) |
67 | 64, 66 | nfeq 2307 |
. . . . . . . 8
⊢
Ⅎ𝑦(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑣〉) |
68 | | opeq2 3743 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑣 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝑣〉) |
69 | 68 | fveq2d 5473 |
. . . . . . . . 9
⊢ (𝑦 = 𝑣 → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑦〉) = (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑣〉)) |
70 | 68 | fveq2d 5473 |
. . . . . . . . 9
⊢ (𝑦 = 𝑣 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑦〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑣〉)) |
71 | 69, 70 | eqeq12d 2172 |
. . . . . . . 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〉))) |
72 | 59, 67, 71 | cbvral 2676 |
. . . . . . 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〉)) |
73 | | opeq1 3742 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑢 → 〈𝑥, 𝑣〉 = 〈𝑢, 𝑣〉) |
74 | 73 | fveq2d 5473 |
. . . . . . . . 9
⊢ (𝑥 = 𝑢 → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑥, 𝑣〉) = (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑢, 𝑣〉)) |
75 | 73 | fveq2d 5473 |
. . . . . . . . 9
⊢ (𝑥 = 𝑢 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑥, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑢, 𝑣〉)) |
76 | 74, 75 | eqeq12d 2172 |
. . . . . . . 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〉))) |
77 | 76 | ralbidv 2457 |
. . . . . . 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〉))) |
78 | 72, 77 | syl5bb 191 |
. . . . . 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〉))) |
79 | 48, 58, 78 | cbvral 2676 |
. . . . 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〉)) |
80 | 47, 79 | sylib 121 |
. . . 4
⊢ (𝜑 → ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑢, 𝑣〉) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑢, 𝑣〉)) |
81 | | fveq2 5469 |
. . . . . 6
⊢ (𝑤 = 〈𝑢, 𝑣〉 → (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = (((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘〈𝑢, 𝑣〉)) |
82 | | fveq2 5469 |
. . . . . 6
⊢ (𝑤 = 〈𝑢, 𝑣〉 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘𝑤) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘〈𝑢, 𝑣〉)) |
83 | 81, 82 | eqeq12d 2172 |
. . . . 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〉))) |
84 | 83 | ralxp 4730 |
. . . 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〉)) |
85 | 80, 84 | sylibr 133 |
. . 3
⊢ (𝜑 → ∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴))‘𝑤) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)‘𝑤)) |
86 | | fco 5336 |
. . . . . 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) |
87 | 31, 11, 86 | syl2anc 409 |
. . . . 5
⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)):(𝑋 × 𝑌)⟶∪ 𝑀) |
88 | 87 | ffnd 5321 |
. . . 4
⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) Fn (𝑋 × 𝑌)) |
89 | 35 | ralrimivva 2539 |
. . . . . 6
⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ ∪ 𝑀) |
90 | 42 | fmpo 6150 |
. . . . . 6
⊢
(∀𝑥 ∈
𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ ∪ 𝑀 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶):(𝑋 × 𝑌)⟶∪ 𝑀) |
91 | 89, 90 | sylib 121 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶):(𝑋 × 𝑌)⟶∪ 𝑀) |
92 | 91 | ffnd 5321 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) Fn (𝑋 × 𝑌)) |
93 | | eqfnfv 5566 |
. . . 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))) |
94 | 88, 92, 93 | syl2anc 409 |
. . 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))) |
95 | 85, 94 | mpbird 166 |
. 2
⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)) |
96 | | cnco 12663 |
. . 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)) |
97 | 9, 25, 96 | syl2anc 409 |
. 2
⊢ (𝜑 → ((𝑧 ∈ 𝑍 ↦ 𝐵) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) |
98 | 95, 97 | eqeltrrd 2235 |
1
⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) |