Proof of Theorem xkocnv
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simprr 772 | . . . . . 6
⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) → 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))) | 
| 2 |  | xkohmeo.x | . . . . . . . . 9
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | 
| 3 | 2 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → 𝐽 ∈ (TopOn‘𝑋)) | 
| 4 |  | xkohmeo.y | . . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) | 
| 5 | 4 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → 𝐾 ∈ (TopOn‘𝑌)) | 
| 6 |  | txtopon 23600 | . . . . . . . . . . . . . 14
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) | 
| 7 | 2, 4, 6 | syl2anc 584 | . . . . . . . . . . . . 13
⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) | 
| 8 | 7 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) | 
| 9 |  | xkohmeo.l | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐿 ∈ Top) | 
| 10 |  | toptopon2 22925 | . . . . . . . . . . . . . 14
⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿)) | 
| 11 | 9, 10 | sylib 218 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿)) | 
| 12 | 11 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → 𝐿 ∈ (TopOn‘∪ 𝐿)) | 
| 13 |  | simpr 484 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) | 
| 14 |  | cnf2 23258 | . . . . . . . . . . . 12
⊢ (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿)
∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → 𝑓:(𝑋 × 𝑌)⟶∪ 𝐿) | 
| 15 | 8, 12, 13, 14 | syl3anc 1372 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → 𝑓:(𝑋 × 𝑌)⟶∪ 𝐿) | 
| 16 | 15 | ffnd 6736 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → 𝑓 Fn (𝑋 × 𝑌)) | 
| 17 |  | fnov 7565 | . . . . . . . . . 10
⊢ (𝑓 Fn (𝑋 × 𝑌) ↔ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) | 
| 18 | 16, 17 | sylib 218 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) | 
| 19 | 18, 13 | eqeltrrd 2841 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) | 
| 20 | 3, 5, 19 | cnmpt2k 23697 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) | 
| 21 | 20 | adantrr 717 | . . . . . 6
⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) | 
| 22 | 1, 21 | eqeltrd 2840 | . . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) → 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) | 
| 23 | 18 | adantrr 717 | . . . . . 6
⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) → 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) | 
| 24 |  | eqid 2736 | . . . . . . 7
⊢ 𝑋 = 𝑋 | 
| 25 |  | nfv 1913 | . . . . . . . . 9
⊢
Ⅎ𝑥𝜑 | 
| 26 |  | nfv 1913 | . . . . . . . . . 10
⊢
Ⅎ𝑥 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) | 
| 27 |  | nfmpt1 5249 | . . . . . . . . . . 11
⊢
Ⅎ𝑥(𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) | 
| 28 | 27 | nfeq2 2922 | . . . . . . . . . 10
⊢
Ⅎ𝑥 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) | 
| 29 | 26, 28 | nfan 1898 | . . . . . . . . 9
⊢
Ⅎ𝑥(𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))) | 
| 30 | 25, 29 | nfan 1898 | . . . . . . . 8
⊢
Ⅎ𝑥(𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) | 
| 31 |  | nfv 1913 | . . . . . . . . . . . . 13
⊢
Ⅎ𝑦𝜑 | 
| 32 |  | nfv 1913 | . . . . . . . . . . . . . 14
⊢
Ⅎ𝑦 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) | 
| 33 |  | nfcv 2904 | . . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑦𝑋 | 
| 34 |  | nfmpt1 5249 | . . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑦(𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)) | 
| 35 | 33, 34 | nfmpt 5248 | . . . . . . . . . . . . . . 15
⊢
Ⅎ𝑦(𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) | 
| 36 | 35 | nfeq2 2922 | . . . . . . . . . . . . . 14
⊢
Ⅎ𝑦 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) | 
| 37 | 32, 36 | nfan 1898 | . . . . . . . . . . . . 13
⊢
Ⅎ𝑦(𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))) | 
| 38 | 31, 37 | nfan 1898 | . . . . . . . . . . . 12
⊢
Ⅎ𝑦(𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) | 
| 39 |  | nfv 1913 | . . . . . . . . . . . 12
⊢
Ⅎ𝑦 𝑥 ∈ 𝑋 | 
| 40 | 38, 39 | nfan 1898 | . . . . . . . . . . 11
⊢
Ⅎ𝑦((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ 𝑥 ∈ 𝑋) | 
| 41 |  | simplrr 777 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))) | 
| 42 | 41 | fveq1d 6907 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑔‘𝑥) = ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))‘𝑥)) | 
| 43 |  | simprl 770 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑥 ∈ 𝑋) | 
| 44 |  | toponmax 22933 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝐾) | 
| 45 | 4, 44 | syl 17 | . . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑌 ∈ 𝐾) | 
| 46 | 45 | ad2antrr 726 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑌 ∈ 𝐾) | 
| 47 | 46 | mptexd 7245 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)) ∈ V) | 
| 48 |  | eqid 2736 | . . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) | 
| 49 | 48 | fvmpt2 7026 | . . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)) ∈ V) → ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))‘𝑥) = (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) | 
| 50 | 43, 47, 49 | syl2anc 584 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))‘𝑥) = (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) | 
| 51 | 42, 50 | eqtrd 2776 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑔‘𝑥) = (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) | 
| 52 | 51 | fveq1d 6907 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑔‘𝑥)‘𝑦) = ((𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))‘𝑦)) | 
| 53 |  | simprr 772 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → 𝑦 ∈ 𝑌) | 
| 54 |  | ovex 7465 | . . . . . . . . . . . . . 14
⊢ (𝑥𝑓𝑦) ∈ V | 
| 55 |  | eqid 2736 | . . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)) = (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)) | 
| 56 | 55 | fvmpt2 7026 | . . . . . . . . . . . . . 14
⊢ ((𝑦 ∈ 𝑌 ∧ (𝑥𝑓𝑦) ∈ V) → ((𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))‘𝑦) = (𝑥𝑓𝑦)) | 
| 57 | 53, 54, 56 | sylancl 586 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))‘𝑦) = (𝑥𝑓𝑦)) | 
| 58 | 52, 57 | eqtrd 2776 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ((𝑔‘𝑥)‘𝑦) = (𝑥𝑓𝑦)) | 
| 59 | 58 | expr 456 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 → ((𝑔‘𝑥)‘𝑦) = (𝑥𝑓𝑦))) | 
| 60 | 40, 59 | ralrimi 3256 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 ((𝑔‘𝑥)‘𝑦) = (𝑥𝑓𝑦)) | 
| 61 |  | eqid 2736 | . . . . . . . . . 10
⊢ 𝑌 = 𝑌 | 
| 62 | 60, 61 | jctil 519 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) ∧ 𝑥 ∈ 𝑋) → (𝑌 = 𝑌 ∧ ∀𝑦 ∈ 𝑌 ((𝑔‘𝑥)‘𝑦) = (𝑥𝑓𝑦))) | 
| 63 | 62 | ex 412 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) → (𝑥 ∈ 𝑋 → (𝑌 = 𝑌 ∧ ∀𝑦 ∈ 𝑌 ((𝑔‘𝑥)‘𝑦) = (𝑥𝑓𝑦)))) | 
| 64 | 30, 63 | ralrimi 3256 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) → ∀𝑥 ∈ 𝑋 (𝑌 = 𝑌 ∧ ∀𝑦 ∈ 𝑌 ((𝑔‘𝑥)‘𝑦) = (𝑥𝑓𝑦))) | 
| 65 |  | mpoeq123 7506 | . . . . . . 7
⊢ ((𝑋 = 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑌 = 𝑌 ∧ ∀𝑦 ∈ 𝑌 ((𝑔‘𝑥)‘𝑦) = (𝑥𝑓𝑦))) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) | 
| 66 | 24, 64, 65 | sylancr 587 | . . . . . 6
⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) | 
| 67 | 23, 66 | eqtr4d 2779 | . . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) → 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))) | 
| 68 | 22, 67 | jca 511 | . . . 4
⊢ ((𝜑 ∧ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) → (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) | 
| 69 |  | simprr 772 | . . . . . 6
⊢ ((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) → 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))) | 
| 70 | 2 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → 𝐽 ∈ (TopOn‘𝑋)) | 
| 71 | 4 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → 𝐾 ∈ (TopOn‘𝑌)) | 
| 72 | 11 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → 𝐿 ∈ (TopOn‘∪ 𝐿)) | 
| 73 |  | xkohmeo.k | . . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally
Comp) | 
| 74 | 73 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → 𝐾 ∈ 𝑛-Locally
Comp) | 
| 75 |  | nllytop 23482 | . . . . . . . . . . . . . 14
⊢ (𝐾 ∈ 𝑛-Locally Comp
→ 𝐾 ∈
Top) | 
| 76 | 74, 75 | syl 17 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → 𝐾 ∈ Top) | 
| 77 | 9 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → 𝐿 ∈ Top) | 
| 78 |  | eqid 2736 | . . . . . . . . . . . . . 14
⊢ (𝐿 ↑ko 𝐾) = (𝐿 ↑ko 𝐾) | 
| 79 | 78 | xkotopon 23609 | . . . . . . . . . . . . 13
⊢ ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿))) | 
| 80 | 76, 77, 79 | syl2anc 584 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿))) | 
| 81 |  | simpr 484 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) | 
| 82 |  | cnf2 23258 | . . . . . . . . . . . 12
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐿 ↑ko 𝐾) ∈ (TopOn‘(𝐾 Cn 𝐿)) ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → 𝑔:𝑋⟶(𝐾 Cn 𝐿)) | 
| 83 | 70, 80, 81, 82 | syl3anc 1372 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → 𝑔:𝑋⟶(𝐾 Cn 𝐿)) | 
| 84 | 83 | feqmptd 6976 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑔‘𝑥))) | 
| 85 | 4 | ad2antrr 726 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌)) | 
| 86 | 11 | ad2antrr 726 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) ∧ 𝑥 ∈ 𝑋) → 𝐿 ∈ (TopOn‘∪ 𝐿)) | 
| 87 | 83 | ffvelcdmda 7103 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) ∧ 𝑥 ∈ 𝑋) → (𝑔‘𝑥) ∈ (𝐾 Cn 𝐿)) | 
| 88 |  | cnf2 23258 | . . . . . . . . . . . . 13
⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿)
∧ (𝑔‘𝑥) ∈ (𝐾 Cn 𝐿)) → (𝑔‘𝑥):𝑌⟶∪ 𝐿) | 
| 89 | 85, 86, 87, 88 | syl3anc 1372 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) ∧ 𝑥 ∈ 𝑋) → (𝑔‘𝑥):𝑌⟶∪ 𝐿) | 
| 90 | 89 | feqmptd 6976 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) ∧ 𝑥 ∈ 𝑋) → (𝑔‘𝑥) = (𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))) | 
| 91 | 90 | mpteq2dva 5241 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → (𝑥 ∈ 𝑋 ↦ (𝑔‘𝑥)) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) | 
| 92 | 84, 91 | eqtrd 2776 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) | 
| 93 | 92, 81 | eqeltrrd 2841 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))) ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) | 
| 94 | 70, 71, 72, 74, 93 | cnmptk2 23695 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾))) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) | 
| 95 | 94 | adantrr 717 | . . . . . 6
⊢ ((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) | 
| 96 | 69, 95 | eqeltrd 2840 | . . . . 5
⊢ ((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) → 𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) | 
| 97 | 92 | adantrr 717 | . . . . . 6
⊢ ((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) → 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) | 
| 98 |  | nfv 1913 | . . . . . . . . 9
⊢
Ⅎ𝑥 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) | 
| 99 |  | nfmpo1 7514 | . . . . . . . . . 10
⊢
Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)) | 
| 100 | 99 | nfeq2 2922 | . . . . . . . . 9
⊢
Ⅎ𝑥 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)) | 
| 101 | 98, 100 | nfan 1898 | . . . . . . . 8
⊢
Ⅎ𝑥(𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))) | 
| 102 | 25, 101 | nfan 1898 | . . . . . . 7
⊢
Ⅎ𝑥(𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) | 
| 103 |  | nfv 1913 | . . . . . . . . . . . 12
⊢
Ⅎ𝑦 𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) | 
| 104 |  | nfmpo2 7515 | . . . . . . . . . . . . 13
⊢
Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)) | 
| 105 | 104 | nfeq2 2922 | . . . . . . . . . . . 12
⊢
Ⅎ𝑦 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)) | 
| 106 | 103, 105 | nfan 1898 | . . . . . . . . . . 11
⊢
Ⅎ𝑦(𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))) | 
| 107 | 31, 106 | nfan 1898 | . . . . . . . . . 10
⊢
Ⅎ𝑦(𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) | 
| 108 | 107, 39 | nfan 1898 | . . . . . . . . 9
⊢
Ⅎ𝑦((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) ∧ 𝑥 ∈ 𝑋) | 
| 109 | 69 | oveqd 7449 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) → (𝑥𝑓𝑦) = (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))𝑦)) | 
| 110 |  | fvex 6918 | . . . . . . . . . . . 12
⊢ ((𝑔‘𝑥)‘𝑦) ∈ V | 
| 111 |  | eqid 2736 | . . . . . . . . . . . . 13
⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)) | 
| 112 | 111 | ovmpt4g 7581 | . . . . . . . . . . . 12
⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ ((𝑔‘𝑥)‘𝑦) ∈ V) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))𝑦) = ((𝑔‘𝑥)‘𝑦)) | 
| 113 | 110, 112 | mp3an3 1451 | . . . . . . . . . . 11
⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))𝑦) = ((𝑔‘𝑥)‘𝑦)) | 
| 114 | 109, 113 | sylan9eq 2796 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝑓𝑦) = ((𝑔‘𝑥)‘𝑦)) | 
| 115 | 114 | expr 456 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 → (𝑥𝑓𝑦) = ((𝑔‘𝑥)‘𝑦))) | 
| 116 | 108, 115 | ralrimi 3256 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 (𝑥𝑓𝑦) = ((𝑔‘𝑥)‘𝑦)) | 
| 117 |  | mpteq12 5233 | . . . . . . . 8
⊢ ((𝑌 = 𝑌 ∧ ∀𝑦 ∈ 𝑌 (𝑥𝑓𝑦) = ((𝑔‘𝑥)‘𝑦)) → (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)) = (𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))) | 
| 118 | 61, 116, 117 | sylancr 587 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)) = (𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))) | 
| 119 | 102, 118 | mpteq2da 5239 | . . . . . 6
⊢ ((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) → (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) | 
| 120 | 97, 119 | eqtr4d 2779 | . . . . 5
⊢ ((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) → 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))) | 
| 121 | 96, 120 | jca 511 | . . . 4
⊢ ((𝜑 ∧ (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) → (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))) | 
| 122 | 68, 121 | impbida 800 | . . 3
⊢ (𝜑 → ((𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))) ↔ (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))))) | 
| 123 | 122 | opabbidv 5208 | . 2
⊢ (𝜑 → {〈𝑔, 𝑓〉 ∣ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))} = {〈𝑔, 𝑓〉 ∣ (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))}) | 
| 124 |  | xkohmeo.f | . . . . 5
⊢ 𝐹 = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))) | 
| 125 |  | df-mpt 5225 | . . . . 5
⊢ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦)))) = {〈𝑓, 𝑔〉 ∣ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))} | 
| 126 | 124, 125 | eqtri 2764 | . . . 4
⊢ 𝐹 = {〈𝑓, 𝑔〉 ∣ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))} | 
| 127 | 126 | cnveqi 5884 | . . 3
⊢ ◡𝐹 = ◡{〈𝑓, 𝑔〉 ∣ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))} | 
| 128 |  | cnvopab 6156 | . . 3
⊢ ◡{〈𝑓, 𝑔〉 ∣ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))} = {〈𝑔, 𝑓〉 ∣ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))} | 
| 129 | 127, 128 | eqtri 2764 | . 2
⊢ ◡𝐹 = {〈𝑔, 𝑓〉 ∣ (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ 𝑔 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))} | 
| 130 |  | df-mpt 5225 | . 2
⊢ (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))) = {〈𝑔, 𝑓〉 ∣ (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ∧ 𝑓 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))} | 
| 131 | 123, 129,
130 | 3eqtr4g 2801 | 1
⊢ (𝜑 → ◡𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦)))) |