Proof of Theorem cnmpt12
Step | Hyp | Ref
| Expression |
1 | | cnmptid.j |
. . . . . 6
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
2 | | cnmpt12.k |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) |
3 | | cnmpt11.a |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) |
4 | | cnf2 12616 |
. . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌) |
5 | 1, 2, 3, 4 | syl3anc 1220 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌) |
6 | 5 | fvmptelrn 5620 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) |
7 | | cnmpt12.l |
. . . . . 6
⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) |
8 | | cnmpt1t.b |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) |
9 | | cnf2 12616 |
. . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍) |
10 | 1, 7, 8, 9 | syl3anc 1220 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍) |
11 | 10 | fvmptelrn 5620 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑍) |
12 | 6, 11 | jca 304 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍)) |
13 | | txtopon 12673 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍))) |
14 | 2, 7, 13 | syl2anc 409 |
. . . . . . . . 9
⊢ (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍))) |
15 | | cnmpt12.c |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀)) |
16 | | cntop2 12613 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀) → 𝑀 ∈ Top) |
17 | 15, 16 | syl 14 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ Top) |
18 | | toptopon2 12428 |
. . . . . . . . . 10
⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀)) |
19 | 17, 18 | sylib 121 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀)) |
20 | | cnf2 12616 |
. . . . . . . . 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) |
21 | 14, 19, 15, 20 | syl3anc 1220 |
. . . . . . . 8
⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶):(𝑌 × 𝑍)⟶∪ 𝑀) |
22 | | eqid 2157 |
. . . . . . . . 9
⊢ (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) = (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) |
23 | 22 | fmpo 6149 |
. . . . . . . 8
⊢
(∀𝑦 ∈
𝑌 ∀𝑧 ∈ 𝑍 𝐶 ∈ ∪ 𝑀 ↔ (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶):(𝑌 × 𝑍)⟶∪ 𝑀) |
24 | 21, 23 | sylibr 133 |
. . . . . . 7
⊢ (𝜑 → ∀𝑦 ∈ 𝑌 ∀𝑧 ∈ 𝑍 𝐶 ∈ ∪ 𝑀) |
25 | | r2al 2476 |
. . . . . . 7
⊢
(∀𝑦 ∈
𝑌 ∀𝑧 ∈ 𝑍 𝐶 ∈ ∪ 𝑀 ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀)) |
26 | 24, 25 | sylib 121 |
. . . . . 6
⊢ (𝜑 → ∀𝑦∀𝑧((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀)) |
27 | 26 | adantr 274 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦∀𝑧((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀)) |
28 | | eleq1 2220 |
. . . . . . . 8
⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑌 ↔ 𝐴 ∈ 𝑌)) |
29 | | eleq1 2220 |
. . . . . . . 8
⊢ (𝑧 = 𝐵 → (𝑧 ∈ 𝑍 ↔ 𝐵 ∈ 𝑍)) |
30 | 28, 29 | bi2anan9 596 |
. . . . . . 7
⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ↔ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍))) |
31 | | cnmpt12.d |
. . . . . . . 8
⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → 𝐶 = 𝐷) |
32 | 31 | eleq1d 2226 |
. . . . . . 7
⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → (𝐶 ∈ ∪ 𝑀 ↔ 𝐷 ∈ ∪ 𝑀)) |
33 | 30, 32 | imbi12d 233 |
. . . . . 6
⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → (((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀) ↔ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍) → 𝐷 ∈ ∪ 𝑀))) |
34 | 33 | spc2gv 2803 |
. . . . 5
⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍) → (∀𝑦∀𝑧((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀) → ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍) → 𝐷 ∈ ∪ 𝑀))) |
35 | 12, 27, 12, 34 | syl3c 63 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ ∪ 𝑀) |
36 | 31, 22 | ovmpoga 5950 |
. . . 4
⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍 ∧ 𝐷 ∈ ∪ 𝑀) → (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵) = 𝐷) |
37 | 6, 11, 35, 36 | syl3anc 1220 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵) = 𝐷) |
38 | 37 | mpteq2dva 4054 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐷)) |
39 | 1, 3, 8, 15 | cnmpt12f 12697 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵)) ∈ (𝐽 Cn 𝑀)) |
40 | 38, 39 | eqeltrrd 2235 |
1
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐷) ∈ (𝐽 Cn 𝑀)) |