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 23258 | . . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌) | 
| 5 | 1, 2, 3, 4 | syl3anc 1372 | . . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌) | 
| 6 | 5 | fvmptelcdm 7132 | . . . 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 23258 | . . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍) | 
| 10 | 1, 7, 8, 9 | syl3anc 1372 | . . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶𝑍) | 
| 11 | 10 | fvmptelcdm 7132 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑍) | 
| 12 | 6, 11 | jca 511 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍)) | 
| 13 |  | txtopon 23600 | . . . . . . . . . 10
⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍))) | 
| 14 | 2, 7, 13 | syl2anc 584 | . . . . . . . . 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 23250 | . . . . . . . . . . 11
⊢ ((𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀) → 𝑀 ∈ Top) | 
| 17 | 15, 16 | syl 17 | . . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ Top) | 
| 18 |  | toptopon2 22925 | . . . . . . . . . 10
⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀)) | 
| 19 | 17, 18 | sylib 218 | . . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀)) | 
| 20 |  | cnf2 23258 | . . . . . . . . 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 1372 | . . . . . . . 8
⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶):(𝑌 × 𝑍)⟶∪ 𝑀) | 
| 22 |  | eqid 2736 | . . . . . . . . 9
⊢ (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) = (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) | 
| 23 | 22 | fmpo 8094 | . . . . . . . 8
⊢
(∀𝑦 ∈
𝑌 ∀𝑧 ∈ 𝑍 𝐶 ∈ ∪ 𝑀 ↔ (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶):(𝑌 × 𝑍)⟶∪ 𝑀) | 
| 24 | 21, 23 | sylibr 234 | . . . . . . 7
⊢ (𝜑 → ∀𝑦 ∈ 𝑌 ∀𝑧 ∈ 𝑍 𝐶 ∈ ∪ 𝑀) | 
| 25 |  | r2al 3194 | . . . . . . 7
⊢
(∀𝑦 ∈
𝑌 ∀𝑧 ∈ 𝑍 𝐶 ∈ ∪ 𝑀 ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀)) | 
| 26 | 24, 25 | sylib 218 | . . . . . 6
⊢ (𝜑 → ∀𝑦∀𝑧((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀)) | 
| 27 | 26 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦∀𝑧((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀)) | 
| 28 |  | eleq1 2828 | . . . . . . . 8
⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑌 ↔ 𝐴 ∈ 𝑌)) | 
| 29 |  | eleq1 2828 | . . . . . . . 8
⊢ (𝑧 = 𝐵 → (𝑧 ∈ 𝑍 ↔ 𝐵 ∈ 𝑍)) | 
| 30 | 28, 29 | bi2anan9 638 | . . . . . . 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 2825 | . . . . . . 7
⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → (𝐶 ∈ ∪ 𝑀 ↔ 𝐷 ∈ ∪ 𝑀)) | 
| 33 | 30, 32 | imbi12d 344 | . . . . . 6
⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → (((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝐶 ∈ ∪ 𝑀) ↔ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍) → 𝐷 ∈ ∪ 𝑀))) | 
| 34 | 33 | spc2gv 3599 | . . . . 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 66 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ ∪ 𝑀) | 
| 36 | 31, 22 | ovmpoga 7588 | . . . 4
⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑍 ∧ 𝐷 ∈ ∪ 𝑀) → (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵) = 𝐷) | 
| 37 | 6, 11, 35, 36 | syl3anc 1372 | . . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵) = 𝐷) | 
| 38 | 37 | mpteq2dva 5241 | . 2
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐷)) | 
| 39 | 1, 3, 8, 15 | cnmpt12f 23675 | . 2
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶)𝐵)) ∈ (𝐽 Cn 𝑀)) | 
| 40 | 38, 39 | eqeltrrd 2841 | 1
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐷) ∈ (𝐽 Cn 𝑀)) |