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Theorem infpwfien 10102
Description: Any infinite well-orderable set is equinumerous to its set of finite subsets. (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
infpwfien ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴)

Proof of Theorem infpwfien
Dummy variables 𝑚 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 infxpidm2 10057 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
2 infn0 9340 . . . . . . . 8 (ω ≼ 𝐴𝐴 ≠ ∅)
32adantl 481 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≠ ∅)
4 fseqen 10067 . . . . . . 7 (((𝐴 × 𝐴) ≈ 𝐴𝐴 ≠ ∅) → 𝑛 ∈ ω (𝐴m 𝑛) ≈ (ω × 𝐴))
51, 3, 4syl2anc 584 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝑛 ∈ ω (𝐴m 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)
86, 1, 7syl2anc 584 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (ω × 𝐴) ≼ 𝐴)
9 endomtr 9052 . . . . . 6 (( 𝑛 ∈ ω (𝐴m 𝑛) ≈ (ω × 𝐴) ∧ (ω × 𝐴) ≼ 𝐴) → 𝑛 ∈ ω (𝐴m 𝑛) ≼ 𝐴)
105, 8, 9syl2anc 584 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝑛 ∈ ω (𝐴m 𝑛) ≼ 𝐴)
11 numdom 10078 . . . . 5 ((𝐴 ∈ dom card ∧ 𝑛 ∈ ω (𝐴m 𝑛) ≼ 𝐴) → 𝑛 ∈ ω (𝐴m 𝑛) ∈ dom card)
1210, 11syldan 591 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝑛 ∈ ω (𝐴m 𝑛) ∈ dom card)
13 eliun 4995 . . . . . . . . 9 (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↔ ∃𝑛 ∈ ω 𝑥 ∈ (𝐴m 𝑛))
14 elmapi 8889 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝐴m 𝑛) → 𝑥:𝑛𝐴)
1514ad2antll 729 . . . . . . . . . . . . . 14 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴m 𝑛))) → 𝑥:𝑛𝐴)
1615frnd 6744 . . . . . . . . . . . . 13 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴m 𝑛))) → ran 𝑥𝐴)
17 vex 3484 . . . . . . . . . . . . . . 15 𝑥 ∈ V
1817rnex 7932 . . . . . . . . . . . . . 14 ran 𝑥 ∈ V
1918elpw 4604 . . . . . . . . . . . . 13 (ran 𝑥 ∈ 𝒫 𝐴 ↔ ran 𝑥𝐴)
2016, 19sylibr 234 . . . . . . . . . . . 12 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴m 𝑛))) → ran 𝑥 ∈ 𝒫 𝐴)
21 simprl 771 . . . . . . . . . . . . . 14 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴m 𝑛))) → 𝑛 ∈ ω)
22 ssid 4006 . . . . . . . . . . . . . 14 𝑛𝑛
23 ssnnfi 9209 . . . . . . . . . . . . . 14 ((𝑛 ∈ ω ∧ 𝑛𝑛) → 𝑛 ∈ Fin)
2421, 22, 23sylancl 586 . . . . . . . . . . . . 13 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴m 𝑛))) → 𝑛 ∈ Fin)
25 ffn 6736 . . . . . . . . . . . . . . 15 (𝑥:𝑛𝐴𝑥 Fn 𝑛)
26 dffn4 6826 . . . . . . . . . . . . . . 15 (𝑥 Fn 𝑛𝑥:𝑛onto→ran 𝑥)
2725, 26sylib 218 . . . . . . . . . . . . . 14 (𝑥:𝑛𝐴𝑥:𝑛onto→ran 𝑥)
2815, 27syl 17 . . . . . . . . . . . . 13 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴m 𝑛))) → 𝑥:𝑛onto→ran 𝑥)
29 fofi 9351 . . . . . . . . . . . . 13 ((𝑛 ∈ Fin ∧ 𝑥:𝑛onto→ran 𝑥) → ran 𝑥 ∈ Fin)
3024, 28, 29syl2anc 584 . . . . . . . . . . . 12 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴m 𝑛))) → ran 𝑥 ∈ Fin)
3120, 30elind 4200 . . . . . . . . . . 11 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴m 𝑛))) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin))
3231expr 456 . . . . . . . . . 10 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → (𝑥 ∈ (𝐴m 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin)))
3332rexlimdva 3155 . . . . . . . . 9 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (∃𝑛 ∈ ω 𝑥 ∈ (𝐴m 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin)))
3413, 33biimtrid 242 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 𝑛 ∈ ω (𝐴m 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin)))
3534imp 406 . . . . . . 7 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑥 𝑛 ∈ ω (𝐴m 𝑛)) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin))
3635fmpttd 7135 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥): 𝑛 ∈ ω (𝐴m 𝑛)⟶(𝒫 𝐴 ∩ Fin))
3736ffnd 6737 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥) Fn 𝑛 ∈ ω (𝐴m 𝑛))
3836frnd 6744 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ran (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥) ⊆ (𝒫 𝐴 ∩ Fin))
39 simpr 484 . . . . . . . . . . . 12 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
4039elin2d 4205 . . . . . . . . . . 11 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ Fin)
41 isfi 9016 . . . . . . . . . . 11 (𝑦 ∈ Fin ↔ ∃𝑚 ∈ ω 𝑦𝑚)
4240, 41sylib 218 . . . . . . . . . 10 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑚 ∈ ω 𝑦𝑚)
43 ensym 9043 . . . . . . . . . . . . 13 (𝑦𝑚𝑚𝑦)
44 bren 8995 . . . . . . . . . . . . 13 (𝑚𝑦 ↔ ∃𝑥 𝑥:𝑚1-1-onto𝑦)
4543, 44sylib 218 . . . . . . . . . . . 12 (𝑦𝑚 → ∃𝑥 𝑥:𝑚1-1-onto𝑦)
46 simprl 771 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑚 ∈ ω)
47 f1of 6848 . . . . . . . . . . . . . . . . . . . 20 (𝑥:𝑚1-1-onto𝑦𝑥:𝑚𝑦)
4847ad2antll 729 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑥:𝑚𝑦)
49 simplr 769 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
5049elin1d 4204 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑦 ∈ 𝒫 𝐴)
5150elpwid 4609 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑦𝐴)
5248, 51fssd 6753 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑥:𝑚𝐴)
53 simplll 775 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝐴 ∈ dom card)
54 vex 3484 . . . . . . . . . . . . . . . . . . 19 𝑚 ∈ V
55 elmapg 8879 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ dom card ∧ 𝑚 ∈ V) → (𝑥 ∈ (𝐴m 𝑚) ↔ 𝑥:𝑚𝐴))
5653, 54, 55sylancl 586 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → (𝑥 ∈ (𝐴m 𝑚) ↔ 𝑥:𝑚𝐴))
5752, 56mpbird 257 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑥 ∈ (𝐴m 𝑚))
58 oveq2 7439 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑚 → (𝐴m 𝑛) = (𝐴m 𝑚))
5958eleq2d 2827 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (𝑥 ∈ (𝐴m 𝑛) ↔ 𝑥 ∈ (𝐴m 𝑚)))
6059rspcev 3622 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ ω ∧ 𝑥 ∈ (𝐴m 𝑚)) → ∃𝑛 ∈ ω 𝑥 ∈ (𝐴m 𝑛))
6146, 57, 60syl2anc 584 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → ∃𝑛 ∈ ω 𝑥 ∈ (𝐴m 𝑛))
6261, 13sylibr 234 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑥 𝑛 ∈ ω (𝐴m 𝑛))
63 f1ofo 6855 . . . . . . . . . . . . . . . . . 18 (𝑥:𝑚1-1-onto𝑦𝑥:𝑚onto𝑦)
6463ad2antll 729 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑥:𝑚onto𝑦)
65 forn 6823 . . . . . . . . . . . . . . . . 17 (𝑥:𝑚onto𝑦 → ran 𝑥 = 𝑦)
6664, 65syl 17 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → ran 𝑥 = 𝑦)
6766eqcomd 2743 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑦 = ran 𝑥)
6862, 67jca 511 . . . . . . . . . . . . . 14 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ∧ 𝑦 = ran 𝑥))
6968expr 456 . . . . . . . . . . . . 13 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑚 ∈ ω) → (𝑥:𝑚1-1-onto𝑦 → (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ∧ 𝑦 = ran 𝑥)))
7069eximdv 1917 . . . . . . . . . . . 12 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑚 ∈ ω) → (∃𝑥 𝑥:𝑚1-1-onto𝑦 → ∃𝑥(𝑥 𝑛 ∈ ω (𝐴m 𝑛) ∧ 𝑦 = ran 𝑥)))
7145, 70syl5 34 . . . . . . . . . . 11 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑚 ∈ ω) → (𝑦𝑚 → ∃𝑥(𝑥 𝑛 ∈ ω (𝐴m 𝑛) ∧ 𝑦 = ran 𝑥)))
7271rexlimdva 3155 . . . . . . . . . 10 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (∃𝑚 ∈ ω 𝑦𝑚 → ∃𝑥(𝑥 𝑛 ∈ ω (𝐴m 𝑛) ∧ 𝑦 = ran 𝑥)))
7342, 72mpd 15 . . . . . . . . 9 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑥(𝑥 𝑛 ∈ ω (𝐴m 𝑛) ∧ 𝑦 = ran 𝑥))
7473ex 412 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → ∃𝑥(𝑥 𝑛 ∈ ω (𝐴m 𝑛) ∧ 𝑦 = ran 𝑥)))
75 eqid 2737 . . . . . . . . . . 11 (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥) = (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥)
7675elrnmpt 5969 . . . . . . . . . 10 (𝑦 ∈ V → (𝑦 ∈ ran (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥) ↔ ∃𝑥 𝑛 ∈ ω (𝐴m 𝑛)𝑦 = ran 𝑥))
7776elv 3485 . . . . . . . . 9 (𝑦 ∈ ran (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥) ↔ ∃𝑥 𝑛 ∈ ω (𝐴m 𝑛)𝑦 = ran 𝑥)
78 df-rex 3071 . . . . . . . . 9 (∃𝑥 𝑛 ∈ ω (𝐴m 𝑛)𝑦 = ran 𝑥 ↔ ∃𝑥(𝑥 𝑛 ∈ ω (𝐴m 𝑛) ∧ 𝑦 = ran 𝑥))
7977, 78bitri 275 . . . . . . . 8 (𝑦 ∈ ran (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥) ↔ ∃𝑥(𝑥 𝑛 ∈ ω (𝐴m 𝑛) ∧ 𝑦 = ran 𝑥))
8074, 79imbitrrdi 252 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ∈ ran (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥)))
8180ssrdv 3989 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ⊆ ran (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥))
8238, 81eqssd 4001 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ran (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥) = (𝒫 𝐴 ∩ Fin))
83 df-fo 6567 . . . . 5 ((𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥): 𝑛 ∈ ω (𝐴m 𝑛)–onto→(𝒫 𝐴 ∩ Fin) ↔ ((𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥) Fn 𝑛 ∈ ω (𝐴m 𝑛) ∧ ran (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥) = (𝒫 𝐴 ∩ Fin)))
8437, 82, 83sylanbrc 583 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥): 𝑛 ∈ ω (𝐴m 𝑛)–onto→(𝒫 𝐴 ∩ Fin))
85 fodomnum 10097 . . . 4 ( 𝑛 ∈ ω (𝐴m 𝑛) ∈ dom card → ((𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥): 𝑛 ∈ ω (𝐴m 𝑛)–onto→(𝒫 𝐴 ∩ Fin) → (𝒫 𝐴 ∩ Fin) ≼ 𝑛 ∈ ω (𝐴m 𝑛)))
8612, 84, 85sylc 65 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≼ 𝑛 ∈ ω (𝐴m 𝑛))
87 domtr 9047 . . 3 (((𝒫 𝐴 ∩ Fin) ≼ 𝑛 ∈ ω (𝐴m 𝑛) ∧ 𝑛 ∈ ω (𝐴m 𝑛) ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≼ 𝐴)
8886, 10, 87syl2anc 584 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≼ 𝐴)
89 pwexg 5378 . . . . 5 (𝐴 ∈ dom card → 𝒫 𝐴 ∈ V)
9089adantr 480 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝒫 𝐴 ∈ V)
91 inex1g 5319 . . . 4 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
9290, 91syl 17 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ∈ V)
93 infpwfidom 10068 . . 3 ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
9492, 93syl 17 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
95 sbth 9133 . 2 (((𝒫 𝐴 ∩ Fin) ≼ 𝐴𝐴 ≼ (𝒫 𝐴 ∩ Fin)) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴)
9688, 94, 95syl2anc 584 1 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wne 2940  wrex 3070  Vcvv 3480  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600   ciun 4991   class class class wbr 5143  cmpt 5225   × cxp 5683  dom cdm 5685  ran crn 5686   Fn wfn 6556  wf 6557  ontowfo 6559  1-1-ontowf1o 6560  (class class class)co 7431  ωcom 7887  m cmap 8866  cen 8982  cdom 8983  Fincfn 8985  cardccrd 9975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seqom 8488  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-card 9979  df-acn 9982
This theorem is referenced by:  inffien  10103  isnumbasgrplem3  43117
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