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Theorem infpwfien 10099
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 10054 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
2 infn0 9337 . . . . . . . 8 (ω ≼ 𝐴𝐴 ≠ ∅)
32adantl 481 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≠ ∅)
4 fseqen 10064 . . . . . . 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 9107 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (ω × 𝐴) ≼ (𝐴 × 𝐴))
7 domentr 9051 . . . . . . 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 9050 . . . . . 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 10075 . . . . 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 4999 . . . . . . . . 9 (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↔ ∃𝑛 ∈ ω 𝑥 ∈ (𝐴m 𝑛))
14 elmapi 8887 . . . . . . . . . . . . . . 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 3481 . . . . . . . . . . . . . . 15 𝑥 ∈ V
1817rnex 7932 . . . . . . . . . . . . . 14 ran 𝑥 ∈ V
1918elpw 4608 . . . . . . . . . . . . 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 4017 . . . . . . . . . . . . . 14 𝑛𝑛
23 ssnnfi 9207 . . . . . . . . . . . . . 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 9348 . . . . . . . . . . . . 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 4209 . . . . . . . . . . 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 3152 . . . . . . . . 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 7134 . . . . . 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 4214 . . . . . . . . . . 11 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ Fin)
41 isfi 9014 . . . . . . . . . . 11 (𝑦 ∈ Fin ↔ ∃𝑚 ∈ ω 𝑦𝑚)
4240, 41sylib 218 . . . . . . . . . 10 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑚 ∈ ω 𝑦𝑚)
43 ensym 9041 . . . . . . . . . . . . 13 (𝑦𝑚𝑚𝑦)
44 bren 8993 . . . . . . . . . . . . 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 4213 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑦 ∈ 𝒫 𝐴)
5150elpwid 4613 . . . . . . . . . . . . . . . . . . 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 3481 . . . . . . . . . . . . . . . . . . 19 𝑚 ∈ V
55 elmapg 8877 . . . . . . . . . . . . . . . . . . 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 7438 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑚 → (𝐴m 𝑛) = (𝐴m 𝑚))
5958eleq2d 2824 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (𝑥 ∈ (𝐴m 𝑛) ↔ 𝑥 ∈ (𝐴m 𝑚)))
6059rspcev 3621 . . . . . . . . . . . . . . . . 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 2740 . . . . . . . . . . . . . . 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 1914 . . . . . . . . . . . 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 3152 . . . . . . . . . 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 2734 . . . . . . . . . . 11 (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥) = (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥)
7675elrnmpt 5971 . . . . . . . . . 10 (𝑦 ∈ V → (𝑦 ∈ ran (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥) ↔ ∃𝑥 𝑛 ∈ ω (𝐴m 𝑛)𝑦 = ran 𝑥))
7776elv 3482 . . . . . . . . 9 (𝑦 ∈ ran (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥) ↔ ∃𝑥 𝑛 ∈ ω (𝐴m 𝑛)𝑦 = ran 𝑥)
78 df-rex 3068 . . . . . . . . 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 4000 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ⊆ ran (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥))
8238, 81eqssd 4012 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ran (𝑥 𝑛 ∈ ω (𝐴m 𝑛) ↦ ran 𝑥) = (𝒫 𝐴 ∩ Fin))
83 df-fo 6568 . . . . 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 10094 . . . 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 9045 . . 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 5383 . . . . 5 (𝐴 ∈ dom card → 𝒫 𝐴 ∈ V)
9089adantr 480 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝒫 𝐴 ∈ V)
91 inex1g 5324 . . . 4 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
9290, 91syl 17 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ∈ V)
93 infpwfidom 10065 . . 3 ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
9492, 93syl 17 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
95 sbth 9131 . 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 1536  wex 1775  wcel 2105  wne 2937  wrex 3067  Vcvv 3477  cin 3961  wss 3962  c0 4338  𝒫 cpw 4604   ciun 4995   class class class wbr 5147  cmpt 5230   × cxp 5686  dom cdm 5688  ran crn 5689   Fn wfn 6557  wf 6558  ontowfo 6560  1-1-ontowf1o 6561  (class class class)co 7430  ωcom 7886  m cmap 8864  cen 8980  cdom 8981  Fincfn 8983  cardccrd 9972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-seqom 8486  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-oi 9547  df-card 9976  df-acn 9979
This theorem is referenced by:  inffien  10100  isnumbasgrplem3  43093
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