Proof of Theorem cnmpt12
Step | Hyp | Ref
| Expression |
1 | | cnmptid.j |
. . . . . . 7
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
2 | | cnmpt12.k |
. . . . . . 7
⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) |
3 | | cnmpt11.a |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) |
4 | | cnf2 21053 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌) |
5 | 1, 2, 3, 4 | syl3anc 1326 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌) |
6 | | eqid 2622 |
. . . . . . 7
⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴) |
7 | 6 | fmpt 6381 |
. . . . . 6
⊢
(∀𝑥 ∈
𝑋 𝐴 ∈ 𝑌 ↔ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌) |
8 | 5, 7 | sylibr 224 |
. . . . 5
⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌) |
9 | 8 | r19.21bi 2932 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) |
10 | | cnmpt12.l |
. . . . . . 7
⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) |
11 | | cnmpt1t.b |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) |
12 | | cnf2 21053 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍) |
13 | 1, 10, 11, 12 | syl3anc 1326 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍) |
14 | | eqid 2622 |
. . . . . . 7
⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵) |
15 | 14 | fmpt 6381 |
. . . . . 6
⊢
(∀𝑥 ∈
𝑋 𝐵 ∈ 𝑍 ↔ (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍) |
16 | 13, 15 | sylibr 224 |
. . . . 5
⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑍) |
17 | 16 | r19.21bi 2932 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑍) |
18 | 9, 17 | jca 554 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍)) |
19 | | txtopon 21394 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍))) |
20 | 2, 10, 19 | syl2anc 693 |
. . . . . . . . 9
⊢ (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍))) |
21 | | cnmpt12.c |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀)) |
22 | | cntop2 21045 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀) → 𝑀 ∈ Top) |
23 | 21, 22 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ Top) |
24 | | eqid 2622 |
. . . . . . . . . . 11
⊢ ∪ 𝑀 =
∪ 𝑀 |
25 | 24 | toptopon 20722 |
. . . . . . . . . 10
⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀)) |
26 | 23, 25 | sylib 208 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀)) |
27 | | cnf2 21053 |
. . . . . . . . 9
⊢ (((𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)) ∧ 𝑀 ∈ (TopOn‘∪ 𝑀)
∧ (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀)) → (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶):(𝑌 × 𝑍)⟶∪ 𝑀) |
28 | 20, 26, 21, 27 | syl3anc 1326 |
. . . . . . . 8
⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶):(𝑌 × 𝑍)⟶∪ 𝑀) |
29 | | eqid 2622 |
. . . . . . . . 9
⊢ (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) = (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) |
30 | 29 | fmpt2 7237 |
. . . . . . . 8
⊢
(∀𝑦 ∈
𝑌 ∀𝑧 ∈ 𝑍 𝐶 ∈ ∪ 𝑀 ↔ (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶):(𝑌 × 𝑍)⟶∪ 𝑀) |
31 | 28, 30 | sylibr 224 |
. . . . . . 7
⊢ (𝜑 → ∀𝑦 ∈ 𝑌 ∀𝑧 ∈ 𝑍 𝐶 ∈ ∪ 𝑀) |
32 | | r2al 2939 |
. . . . . . 7
⊢
(∀𝑦 ∈
𝑌 ∀𝑧 ∈ 𝑍 𝐶 ∈ ∪ 𝑀 ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀)) |
33 | 31, 32 | sylib 208 |
. . . . . 6
⊢ (𝜑 → ∀𝑦∀𝑧((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀)) |
34 | 33 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦∀𝑧((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀)) |
35 | | eleq1 2689 |
. . . . . . . 8
⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑌 ↔ 𝐴 ∈ 𝑌)) |
36 | | eleq1 2689 |
. . . . . . . 8
⊢ (𝑧 = 𝐵 → (𝑧 ∈ 𝑍 ↔ 𝐵 ∈ 𝑍)) |
37 | 35, 36 | bi2anan9 917 |
. . . . . . 7
⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ↔ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍))) |
38 | | cnmpt12.d |
. . . . . . . 8
⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → 𝐶 = 𝐷) |
39 | 38 | eleq1d 2686 |
. . . . . . 7
⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → (𝐶 ∈ ∪ 𝑀 ↔ 𝐷 ∈ ∪ 𝑀)) |
40 | 37, 39 | imbi12d 334 |
. . . . . 6
⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → (((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀) ↔ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍) → 𝐷 ∈ ∪ 𝑀))) |
41 | 40 | spc2gv 3296 |
. . . . 5
⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍) → (∀𝑦∀𝑧((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀) → ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍) → 𝐷 ∈ ∪ 𝑀))) |
42 | 18, 34, 18, 41 | syl3c 66 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ ∪ 𝑀) |
43 | 38, 29 | ovmpt2ga 6790 |
. . . 4
⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍 ∧ 𝐷 ∈ ∪ 𝑀) → (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵) = 𝐷) |
44 | 9, 17, 42, 43 | syl3anc 1326 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵) = 𝐷) |
45 | 44 | mpteq2dva 4744 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐷)) |
46 | 1, 3, 11, 21 | cnmpt12f 21469 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵)) ∈ (𝐽 Cn 𝑀)) |
47 | 45, 46 | eqeltrrd 2702 |
1
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐷) ∈ (𝐽 Cn 𝑀)) |