Proof of Theorem elpt
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | ptbas.1 | . . 3
⊢ 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} | 
| 2 | 1 | eleq2i 2833 | . 2
⊢ (𝑆 ∈ 𝐵 ↔ 𝑆 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}) | 
| 3 |  | simpr 484 | . . . . 5
⊢ (((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) → 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) | 
| 4 |  | ixpexg 8962 | . . . . . 6
⊢
(∀𝑦 ∈
𝐴 (𝑔‘𝑦) ∈ V → X𝑦 ∈
𝐴 (𝑔‘𝑦) ∈ V) | 
| 5 |  | fvexd 6921 | . . . . . 6
⊢ (𝑦 ∈ 𝐴 → (𝑔‘𝑦) ∈ V) | 
| 6 | 4, 5 | mprg 3067 | . . . . 5
⊢ X𝑦 ∈
𝐴 (𝑔‘𝑦) ∈ V | 
| 7 | 3, 6 | eqeltrdi 2849 | . . . 4
⊢ (((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) → 𝑆 ∈ V) | 
| 8 | 7 | exlimiv 1930 | . . 3
⊢
(∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) → 𝑆 ∈ V) | 
| 9 |  | eqeq1 2741 | . . . . 5
⊢ (𝑥 = 𝑆 → (𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) ↔ 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))) | 
| 10 | 9 | anbi2d 630 | . . . 4
⊢ (𝑥 = 𝑆 → (((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) ↔ ((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)))) | 
| 11 | 10 | exbidv 1921 | . . 3
⊢ (𝑥 = 𝑆 → (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) ↔ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)))) | 
| 12 | 8, 11 | elab3 3686 | . 2
⊢ (𝑆 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} ↔ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))) | 
| 13 |  | fneq1 6659 | . . . . 5
⊢ (𝑔 = ℎ → (𝑔 Fn 𝐴 ↔ ℎ Fn 𝐴)) | 
| 14 |  | fveq1 6905 | . . . . . . 7
⊢ (𝑔 = ℎ → (𝑔‘𝑦) = (ℎ‘𝑦)) | 
| 15 | 14 | eleq1d 2826 | . . . . . 6
⊢ (𝑔 = ℎ → ((𝑔‘𝑦) ∈ (𝐹‘𝑦) ↔ (ℎ‘𝑦) ∈ (𝐹‘𝑦))) | 
| 16 | 15 | ralbidv 3178 | . . . . 5
⊢ (𝑔 = ℎ → (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦))) | 
| 17 | 14 | eqeq1d 2739 | . . . . . . 7
⊢ (𝑔 = ℎ → ((𝑔‘𝑦) = ∪ (𝐹‘𝑦) ↔ (ℎ‘𝑦) = ∪ (𝐹‘𝑦))) | 
| 18 | 17 | rexralbidv 3223 | . . . . . 6
⊢ (𝑔 = ℎ → (∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) | 
| 19 |  | difeq2 4120 | . . . . . . . 8
⊢ (𝑧 = 𝑤 → (𝐴 ∖ 𝑧) = (𝐴 ∖ 𝑤)) | 
| 20 | 19 | raleqdv 3326 | . . . . . . 7
⊢ (𝑧 = 𝑤 → (∀𝑦 ∈ (𝐴 ∖ 𝑧)(ℎ‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) | 
| 21 | 20 | cbvrexvw 3238 | . . . . . 6
⊢
(∃𝑧 ∈ Fin
∀𝑦 ∈ (𝐴 ∖ 𝑧)(ℎ‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)) | 
| 22 | 18, 21 | bitrdi 287 | . . . . 5
⊢ (𝑔 = ℎ → (∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦) ↔ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦))) | 
| 23 | 13, 16, 22 | 3anbi123d 1438 | . . . 4
⊢ (𝑔 = ℎ → ((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ↔ (ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)))) | 
| 24 | 14 | ixpeq2dv 8953 | . . . . 5
⊢ (𝑔 = ℎ → X𝑦 ∈ 𝐴 (𝑔‘𝑦) = X𝑦 ∈ 𝐴 (ℎ‘𝑦)) | 
| 25 | 24 | eqeq2d 2748 | . . . 4
⊢ (𝑔 = ℎ → (𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) ↔ 𝑆 = X𝑦 ∈ 𝐴 (ℎ‘𝑦))) | 
| 26 | 23, 25 | anbi12d 632 | . . 3
⊢ (𝑔 = ℎ → (((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) ↔ ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (ℎ‘𝑦)))) | 
| 27 | 26 | cbvexvw 2036 | . 2
⊢
(∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) ↔ ∃ℎ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (ℎ‘𝑦))) | 
| 28 | 2, 12, 27 | 3bitri 297 | 1
⊢ (𝑆 ∈ 𝐵 ↔ ∃ℎ((ℎ Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑤)(ℎ‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑆 = X𝑦 ∈ 𝐴 (ℎ‘𝑦))) |