| Step | Hyp | Ref
| Expression |
| 1 | | infxpidm2 10057 |
. . . . . . 7
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴) |
| 2 | | infn0 9340 |
. . . . . . . 8
⊢ (ω
≼ 𝐴 → 𝐴 ≠ ∅) |
| 3 | 2 | adantl 481 |
. . . . . . 7
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) → 𝐴 ≠ ∅) |
| 4 | | fseqen 10067 |
. . . . . . 7
⊢ (((𝐴 × 𝐴) ≈ 𝐴 ∧ 𝐴 ≠ ∅) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴)) |
| 5 | 1, 3, 4 | syl2anc 584 |
. . . . . 6
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴)) |
| 6 | | xpdom1g 9109 |
. . . . . . 7
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) → (ω
× 𝐴) ≼ (𝐴 × 𝐴)) |
| 7 | | domentr 9053 |
. . . . . . 7
⊢
(((ω × 𝐴) ≼ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ≈ 𝐴) → (ω × 𝐴) ≼ 𝐴) |
| 8 | 6, 1, 7 | syl2anc 584 |
. . . . . 6
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) → (ω
× 𝐴) ≼ 𝐴) |
| 9 | | endomtr 9052 |
. . . . . 6
⊢
((∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴) ∧ (ω × 𝐴) ≼ 𝐴) → ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ 𝐴) |
| 10 | 5, 8, 9 | syl2anc 584 |
. . . . 5
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ 𝐴) |
| 11 | | numdom 10078 |
. . . . 5
⊢ ((𝐴 ∈ dom card ∧ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ 𝐴) → ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ∈ dom
card) |
| 12 | 10, 11 | syldan 591 |
. . . 4
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∈ dom card) |
| 13 | | eliun 4995 |
. . . . . . . . 9
⊢ (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↔ ∃𝑛 ∈ ω 𝑥 ∈ (𝐴 ↑m 𝑛)) |
| 14 | | elmapi 8889 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (𝐴 ↑m 𝑛) → 𝑥:𝑛⟶𝐴) |
| 15 | 14 | ad2antll 729 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → 𝑥:𝑛⟶𝐴) |
| 16 | 15 | frnd 6744 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → ran 𝑥 ⊆ 𝐴) |
| 17 | | vex 3484 |
. . . . . . . . . . . . . . 15
⊢ 𝑥 ∈ V |
| 18 | 17 | rnex 7932 |
. . . . . . . . . . . . . 14
⊢ ran 𝑥 ∈ V |
| 19 | 18 | elpw 4604 |
. . . . . . . . . . . . 13
⊢ (ran
𝑥 ∈ 𝒫 𝐴 ↔ ran 𝑥 ⊆ 𝐴) |
| 20 | 16, 19 | sylibr 234 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → ran 𝑥 ∈ 𝒫 𝐴) |
| 21 | | simprl 771 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → 𝑛 ∈ ω) |
| 22 | | ssid 4006 |
. . . . . . . . . . . . . 14
⊢ 𝑛 ⊆ 𝑛 |
| 23 | | ssnnfi 9209 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ ω ∧ 𝑛 ⊆ 𝑛) → 𝑛 ∈ Fin) |
| 24 | 21, 22, 23 | sylancl 586 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → 𝑛 ∈ Fin) |
| 25 | | ffn 6736 |
. . . . . . . . . . . . . . 15
⊢ (𝑥:𝑛⟶𝐴 → 𝑥 Fn 𝑛) |
| 26 | | dffn4 6826 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 Fn 𝑛 ↔ 𝑥:𝑛–onto→ran 𝑥) |
| 27 | 25, 26 | sylib 218 |
. . . . . . . . . . . . . 14
⊢ (𝑥:𝑛⟶𝐴 → 𝑥:𝑛–onto→ran 𝑥) |
| 28 | 15, 27 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → 𝑥:𝑛–onto→ran 𝑥) |
| 29 | | fofi 9351 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈ Fin ∧ 𝑥:𝑛–onto→ran 𝑥) → ran 𝑥 ∈ Fin) |
| 30 | 24, 28, 29 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → ran 𝑥 ∈ Fin) |
| 31 | 20, 30 | elind 4200 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑛))) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) |
| 32 | 31 | expr 456 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑛 ∈ ω) → (𝑥 ∈ (𝐴 ↑m 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin))) |
| 33 | 32 | rexlimdva 3155 |
. . . . . . . . 9
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) →
(∃𝑛 ∈ ω
𝑥 ∈ (𝐴 ↑m 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin))) |
| 34 | 13, 33 | biimtrid 242 |
. . . . . . . 8
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) → (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin))) |
| 35 | 34 | imp 406 |
. . . . . . 7
⊢ (((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) |
| 36 | 35 | fmpttd 7135 |
. . . . . 6
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) → (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥):∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)⟶(𝒫 𝐴 ∩ Fin)) |
| 37 | 36 | ffnd 6737 |
. . . . 5
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) → (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) Fn ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛)) |
| 38 | 36 | frnd 6744 |
. . . . . 6
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) → ran
(𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) ⊆ (𝒫 𝐴 ∩ Fin)) |
| 39 | | simpr 484 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) |
| 40 | 39 | elin2d 4205 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ Fin) |
| 41 | | isfi 9016 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ Fin ↔ ∃𝑚 ∈ ω 𝑦 ≈ 𝑚) |
| 42 | 40, 41 | sylib 218 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑚 ∈ ω 𝑦 ≈ 𝑚) |
| 43 | | ensym 9043 |
. . . . . . . . . . . . 13
⊢ (𝑦 ≈ 𝑚 → 𝑚 ≈ 𝑦) |
| 44 | | bren 8995 |
. . . . . . . . . . . . 13
⊢ (𝑚 ≈ 𝑦 ↔ ∃𝑥 𝑥:𝑚–1-1-onto→𝑦) |
| 45 | 43, 44 | sylib 218 |
. . . . . . . . . . . 12
⊢ (𝑦 ≈ 𝑚 → ∃𝑥 𝑥:𝑚–1-1-onto→𝑦) |
| 46 | | simprl 771 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑚 ∈ ω) |
| 47 | | f1of 6848 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥:𝑚–1-1-onto→𝑦 → 𝑥:𝑚⟶𝑦) |
| 48 | 47 | ad2antll 729 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥:𝑚⟶𝑦) |
| 49 | | simplr 769 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) |
| 50 | 49 | elin1d 4204 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑦 ∈ 𝒫 𝐴) |
| 51 | 50 | elpwid 4609 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑦 ⊆ 𝐴) |
| 52 | 48, 51 | fssd 6753 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥:𝑚⟶𝐴) |
| 53 | | simplll 775 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝐴 ∈ dom card) |
| 54 | | vex 3484 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑚 ∈ V |
| 55 | | elmapg 8879 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∈ dom card ∧ 𝑚 ∈ V) → (𝑥 ∈ (𝐴 ↑m 𝑚) ↔ 𝑥:𝑚⟶𝐴)) |
| 56 | 53, 54, 55 | sylancl 586 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → (𝑥 ∈ (𝐴 ↑m 𝑚) ↔ 𝑥:𝑚⟶𝐴)) |
| 57 | 52, 56 | mpbird 257 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥 ∈ (𝐴 ↑m 𝑚)) |
| 58 | | oveq2 7439 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 = 𝑚 → (𝐴 ↑m 𝑛) = (𝐴 ↑m 𝑚)) |
| 59 | 58 | eleq2d 2827 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 𝑚 → (𝑥 ∈ (𝐴 ↑m 𝑛) ↔ 𝑥 ∈ (𝐴 ↑m 𝑚))) |
| 60 | 59 | rspcev 3622 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑚 ∈ ω ∧ 𝑥 ∈ (𝐴 ↑m 𝑚)) → ∃𝑛 ∈ ω 𝑥 ∈ (𝐴 ↑m 𝑛)) |
| 61 | 46, 57, 60 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → ∃𝑛 ∈ ω 𝑥 ∈ (𝐴 ↑m 𝑛)) |
| 62 | 61, 13 | sylibr 234 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛)) |
| 63 | | f1ofo 6855 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥:𝑚–1-1-onto→𝑦 → 𝑥:𝑚–onto→𝑦) |
| 64 | 63 | ad2antll 729 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑥:𝑚–onto→𝑦) |
| 65 | | forn 6823 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥:𝑚–onto→𝑦 → ran 𝑥 = 𝑦) |
| 66 | 64, 65 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → ran 𝑥 = 𝑦) |
| 67 | 66 | eqcomd 2743 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → 𝑦 = ran 𝑥) |
| 68 | 62, 67 | jca 511 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚–1-1-onto→𝑦)) → (𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥)) |
| 69 | 68 | expr 456 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑚 ∈ ω) → (𝑥:𝑚–1-1-onto→𝑦 → (𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥))) |
| 70 | 69 | eximdv 1917 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑚 ∈ ω) →
(∃𝑥 𝑥:𝑚–1-1-onto→𝑦 → ∃𝑥(𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥))) |
| 71 | 45, 70 | syl5 34 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑚 ∈ ω) → (𝑦 ≈ 𝑚 → ∃𝑥(𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥))) |
| 72 | 71 | rexlimdva 3155 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) →
(∃𝑚 ∈ ω
𝑦 ≈ 𝑚 → ∃𝑥(𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥))) |
| 73 | 42, 72 | mpd 15 |
. . . . . . . . 9
⊢ (((𝐴 ∈ dom card ∧ ω
≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑥(𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥)) |
| 74 | 73 | ex 412 |
. . . . . . . 8
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → ∃𝑥(𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥))) |
| 75 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) = (𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) |
| 76 | 75 | elrnmpt 5969 |
. . . . . . . . . 10
⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) ↔ ∃𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛)𝑦 = ran 𝑥)) |
| 77 | 76 | elv 3485 |
. . . . . . . . 9
⊢ (𝑦 ∈ ran (𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) ↔ ∃𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛)𝑦 = ran 𝑥) |
| 78 | | df-rex 3071 |
. . . . . . . . 9
⊢
(∃𝑥 ∈
∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)𝑦 = ran 𝑥 ↔ ∃𝑥(𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥)) |
| 79 | 77, 78 | bitri 275 |
. . . . . . . 8
⊢ (𝑦 ∈ ran (𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) ↔ ∃𝑥(𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ 𝑦 = ran 𝑥)) |
| 80 | 74, 79 | imbitrrdi 252 |
. . . . . . 7
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ∈ ran (𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥))) |
| 81 | 80 | ssrdv 3989 |
. . . . . 6
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) →
(𝒫 𝐴 ∩ Fin)
⊆ ran (𝑥 ∈
∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥)) |
| 82 | 38, 81 | eqssd 4001 |
. . . . 5
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) → ran
(𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) = (𝒫 𝐴 ∩ Fin)) |
| 83 | | df-fo 6567 |
. . . . 5
⊢ ((𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥):∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)–onto→(𝒫 𝐴 ∩ Fin) ↔ ((𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) Fn ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ ran (𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥) = (𝒫 𝐴 ∩ Fin))) |
| 84 | 37, 82, 83 | sylanbrc 583 |
. . . 4
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) → (𝑥 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥):∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)–onto→(𝒫 𝐴 ∩ Fin)) |
| 85 | | fodomnum 10097 |
. . . 4
⊢ (∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∈ dom card → ((𝑥 ∈ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ↦ ran 𝑥):∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)–onto→(𝒫 𝐴 ∩ Fin) → (𝒫 𝐴 ∩ Fin) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))) |
| 86 | 12, 84, 85 | sylc 65 |
. . 3
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) →
(𝒫 𝐴 ∩ Fin)
≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) |
| 87 | | domtr 9047 |
. . 3
⊢
(((𝒫 𝐴 ∩
Fin) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ ∪
𝑛 ∈ ω (𝐴 ↑m 𝑛) ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≼ 𝐴) |
| 88 | 86, 10, 87 | syl2anc 584 |
. 2
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) →
(𝒫 𝐴 ∩ Fin)
≼ 𝐴) |
| 89 | | pwexg 5378 |
. . . . 5
⊢ (𝐴 ∈ dom card →
𝒫 𝐴 ∈
V) |
| 90 | 89 | adantr 480 |
. . . 4
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) → 𝒫
𝐴 ∈
V) |
| 91 | | inex1g 5319 |
. . . 4
⊢
(𝒫 𝐴 ∈
V → (𝒫 𝐴 ∩
Fin) ∈ V) |
| 92 | 90, 91 | syl 17 |
. . 3
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) →
(𝒫 𝐴 ∩ Fin)
∈ V) |
| 93 | | infpwfidom 10068 |
. . 3
⊢
((𝒫 𝐴 ∩
Fin) ∈ V → 𝐴
≼ (𝒫 𝐴 ∩
Fin)) |
| 94 | 92, 93 | syl 17 |
. 2
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) → 𝐴 ≼ (𝒫 𝐴 ∩ Fin)) |
| 95 | | sbth 9133 |
. 2
⊢
(((𝒫 𝐴 ∩
Fin) ≼ 𝐴 ∧ 𝐴 ≼ (𝒫 𝐴 ∩ Fin)) → (𝒫
𝐴 ∩ Fin) ≈ 𝐴) |
| 96 | 88, 94, 95 | syl2anc 584 |
1
⊢ ((𝐴 ∈ dom card ∧ ω
≼ 𝐴) →
(𝒫 𝐴 ∩ Fin)
≈ 𝐴) |