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Theorem infpwfien 8870
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 8825 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
2 infn0 8207 . . . . . . . 8 (ω ≼ 𝐴𝐴 ≠ ∅)
32adantl 482 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≠ ∅)
4 fseqen 8835 . . . . . . 7 (((𝐴 × 𝐴) ≈ 𝐴𝐴 ≠ ∅) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≈ (ω × 𝐴))
51, 3, 4syl2anc 692 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≈ (ω × 𝐴))
6 xpdom1g 8042 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (ω × 𝐴) ≼ (𝐴 × 𝐴))
7 domentr 8000 . . . . . . 7 (((ω × 𝐴) ≼ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ≈ 𝐴) → (ω × 𝐴) ≼ 𝐴)
86, 1, 7syl2anc 692 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (ω × 𝐴) ≼ 𝐴)
9 endomtr 7999 . . . . . 6 (( 𝑛 ∈ ω (𝐴𝑚 𝑛) ≈ (ω × 𝐴) ∧ (ω × 𝐴) ≼ 𝐴) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≼ 𝐴)
105, 8, 9syl2anc 692 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≼ 𝐴)
11 numdom 8846 . . . . 5 ((𝐴 ∈ dom card ∧ 𝑛 ∈ ω (𝐴𝑚 𝑛) ≼ 𝐴) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ∈ dom card)
1210, 11syldan 487 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ∈ dom card)
13 eliun 4515 . . . . . . . . 9 (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↔ ∃𝑛 ∈ ω 𝑥 ∈ (𝐴𝑚 𝑛))
14 elmapi 7864 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝐴𝑚 𝑛) → 𝑥:𝑛𝐴)
1514ad2antll 764 . . . . . . . . . . . . . 14 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴𝑚 𝑛))) → 𝑥:𝑛𝐴)
16 frn 6040 . . . . . . . . . . . . . 14 (𝑥:𝑛𝐴 → ran 𝑥𝐴)
1715, 16syl 17 . . . . . . . . . . . . 13 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴𝑚 𝑛))) → ran 𝑥𝐴)
18 vex 3198 . . . . . . . . . . . . . . 15 𝑥 ∈ V
1918rnex 7085 . . . . . . . . . . . . . 14 ran 𝑥 ∈ V
2019elpw 4155 . . . . . . . . . . . . 13 (ran 𝑥 ∈ 𝒫 𝐴 ↔ ran 𝑥𝐴)
2117, 20sylibr 224 . . . . . . . . . . . 12 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴𝑚 𝑛))) → ran 𝑥 ∈ 𝒫 𝐴)
22 simprl 793 . . . . . . . . . . . . . 14 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴𝑚 𝑛))) → 𝑛 ∈ ω)
23 ssid 3616 . . . . . . . . . . . . . 14 𝑛𝑛
24 ssnnfi 8164 . . . . . . . . . . . . . 14 ((𝑛 ∈ ω ∧ 𝑛𝑛) → 𝑛 ∈ Fin)
2522, 23, 24sylancl 693 . . . . . . . . . . . . 13 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴𝑚 𝑛))) → 𝑛 ∈ Fin)
26 ffn 6032 . . . . . . . . . . . . . . 15 (𝑥:𝑛𝐴𝑥 Fn 𝑛)
27 dffn4 6108 . . . . . . . . . . . . . . 15 (𝑥 Fn 𝑛𝑥:𝑛onto→ran 𝑥)
2826, 27sylib 208 . . . . . . . . . . . . . 14 (𝑥:𝑛𝐴𝑥:𝑛onto→ran 𝑥)
2915, 28syl 17 . . . . . . . . . . . . 13 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴𝑚 𝑛))) → 𝑥:𝑛onto→ran 𝑥)
30 fofi 8237 . . . . . . . . . . . . 13 ((𝑛 ∈ Fin ∧ 𝑥:𝑛onto→ran 𝑥) → ran 𝑥 ∈ Fin)
3125, 29, 30syl2anc 692 . . . . . . . . . . . 12 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴𝑚 𝑛))) → ran 𝑥 ∈ Fin)
3221, 31elind 3790 . . . . . . . . . . 11 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴𝑚 𝑛))) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin))
3332expr 642 . . . . . . . . . 10 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → (𝑥 ∈ (𝐴𝑚 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin)))
3433rexlimdva 3027 . . . . . . . . 9 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (∃𝑛 ∈ ω 𝑥 ∈ (𝐴𝑚 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin)))
3513, 34syl5bi 232 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin)))
3635imp 445 . . . . . . 7 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛)) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin))
37 eqid 2620 . . . . . . 7 (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥) = (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥)
3836, 37fmptd 6371 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥): 𝑛 ∈ ω (𝐴𝑚 𝑛)⟶(𝒫 𝐴 ∩ Fin))
39 ffn 6032 . . . . . 6 ((𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥): 𝑛 ∈ ω (𝐴𝑚 𝑛)⟶(𝒫 𝐴 ∩ Fin) → (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥) Fn 𝑛 ∈ ω (𝐴𝑚 𝑛))
4038, 39syl 17 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥) Fn 𝑛 ∈ ω (𝐴𝑚 𝑛))
41 frn 6040 . . . . . . 7 ((𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥): 𝑛 ∈ ω (𝐴𝑚 𝑛)⟶(𝒫 𝐴 ∩ Fin) → ran (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥) ⊆ (𝒫 𝐴 ∩ Fin))
4238, 41syl 17 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ran (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥) ⊆ (𝒫 𝐴 ∩ Fin))
43 inss2 3826 . . . . . . . . . . . 12 (𝒫 𝐴 ∩ Fin) ⊆ Fin
44 simpr 477 . . . . . . . . . . . 12 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
4543, 44sseldi 3593 . . . . . . . . . . 11 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ Fin)
46 isfi 7964 . . . . . . . . . . 11 (𝑦 ∈ Fin ↔ ∃𝑚 ∈ ω 𝑦𝑚)
4745, 46sylib 208 . . . . . . . . . 10 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑚 ∈ ω 𝑦𝑚)
48 ensym 7990 . . . . . . . . . . . . 13 (𝑦𝑚𝑚𝑦)
49 bren 7949 . . . . . . . . . . . . 13 (𝑚𝑦 ↔ ∃𝑥 𝑥:𝑚1-1-onto𝑦)
5048, 49sylib 208 . . . . . . . . . . . 12 (𝑦𝑚 → ∃𝑥 𝑥:𝑚1-1-onto𝑦)
51 simprl 793 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑚 ∈ ω)
52 f1of 6124 . . . . . . . . . . . . . . . . . . . 20 (𝑥:𝑚1-1-onto𝑦𝑥:𝑚𝑦)
5352ad2antll 764 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑥:𝑚𝑦)
54 inss1 3825 . . . . . . . . . . . . . . . . . . . . 21 (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
55 simplr 791 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
5654, 55sseldi 3593 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑦 ∈ 𝒫 𝐴)
5756elpwid 4161 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑦𝐴)
5853, 57fssd 6044 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑥:𝑚𝐴)
59 simplll 797 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝐴 ∈ dom card)
60 vex 3198 . . . . . . . . . . . . . . . . . . 19 𝑚 ∈ V
61 elmapg 7855 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ dom card ∧ 𝑚 ∈ V) → (𝑥 ∈ (𝐴𝑚 𝑚) ↔ 𝑥:𝑚𝐴))
6259, 60, 61sylancl 693 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → (𝑥 ∈ (𝐴𝑚 𝑚) ↔ 𝑥:𝑚𝐴))
6358, 62mpbird 247 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑥 ∈ (𝐴𝑚 𝑚))
64 oveq2 6643 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑚 → (𝐴𝑚 𝑛) = (𝐴𝑚 𝑚))
6564eleq2d 2685 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (𝑥 ∈ (𝐴𝑚 𝑛) ↔ 𝑥 ∈ (𝐴𝑚 𝑚)))
6665rspcev 3304 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ ω ∧ 𝑥 ∈ (𝐴𝑚 𝑚)) → ∃𝑛 ∈ ω 𝑥 ∈ (𝐴𝑚 𝑛))
6751, 63, 66syl2anc 692 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → ∃𝑛 ∈ ω 𝑥 ∈ (𝐴𝑚 𝑛))
6867, 13sylibr 224 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛))
69 f1ofo 6131 . . . . . . . . . . . . . . . . . 18 (𝑥:𝑚1-1-onto𝑦𝑥:𝑚onto𝑦)
7069ad2antll 764 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑥:𝑚onto𝑦)
71 forn 6105 . . . . . . . . . . . . . . . . 17 (𝑥:𝑚onto𝑦 → ran 𝑥 = 𝑦)
7270, 71syl 17 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → ran 𝑥 = 𝑦)
7372eqcomd 2626 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑦 = ran 𝑥)
7468, 73jca 554 . . . . . . . . . . . . . 14 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ 𝑦 = ran 𝑥))
7574expr 642 . . . . . . . . . . . . 13 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑚 ∈ ω) → (𝑥:𝑚1-1-onto𝑦 → (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ 𝑦 = ran 𝑥)))
7675eximdv 1844 . . . . . . . . . . . 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)))
7750, 76syl5 34 . . . . . . . . . . 11 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑚 ∈ ω) → (𝑦𝑚 → ∃𝑥(𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ 𝑦 = ran 𝑥)))
7877rexlimdva 3027 . . . . . . . . . 10 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (∃𝑚 ∈ ω 𝑦𝑚 → ∃𝑥(𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ 𝑦 = ran 𝑥)))
7947, 78mpd 15 . . . . . . . . 9 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑥(𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ 𝑦 = ran 𝑥))
8079ex 450 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → ∃𝑥(𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ 𝑦 = ran 𝑥)))
81 vex 3198 . . . . . . . . . 10 𝑦 ∈ V
8237elrnmpt 5361 . . . . . . . . . 10 (𝑦 ∈ V → (𝑦 ∈ ran (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥) ↔ ∃𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛)𝑦 = ran 𝑥))
8381, 82ax-mp 5 . . . . . . . . 9 (𝑦 ∈ ran (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥) ↔ ∃𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛)𝑦 = ran 𝑥)
84 df-rex 2915 . . . . . . . . 9 (∃𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛)𝑦 = ran 𝑥 ↔ ∃𝑥(𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ 𝑦 = ran 𝑥))
8583, 84bitri 264 . . . . . . . 8 (𝑦 ∈ ran (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥) ↔ ∃𝑥(𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ 𝑦 = ran 𝑥))
8680, 85syl6ibr 242 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ∈ ran (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥)))
8786ssrdv 3601 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ⊆ ran (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥))
8842, 87eqssd 3612 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ran (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥) = (𝒫 𝐴 ∩ Fin))
89 df-fo 5882 . . . . 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)))
9040, 88, 89sylanbrc 697 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥): 𝑛 ∈ ω (𝐴𝑚 𝑛)–onto→(𝒫 𝐴 ∩ Fin))
91 fodomnum 8865 . . . 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)))
9212, 90, 91sylc 65 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≼ 𝑛 ∈ ω (𝐴𝑚 𝑛))
93 domtr 7994 . . 3 (((𝒫 𝐴 ∩ Fin) ≼ 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ 𝑛 ∈ ω (𝐴𝑚 𝑛) ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≼ 𝐴)
9492, 10, 93syl2anc 692 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≼ 𝐴)
95 pwexg 4841 . . . . 5 (𝐴 ∈ dom card → 𝒫 𝐴 ∈ V)
9695adantr 481 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝒫 𝐴 ∈ V)
97 inex1g 4792 . . . 4 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
9896, 97syl 17 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ∈ V)
99 infpwfidom 8836 . . 3 ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
10098, 99syl 17 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
101 sbth 8065 . 2 (((𝒫 𝐴 ∩ Fin) ≼ 𝐴𝐴 ≼ (𝒫 𝐴 ∩ Fin)) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴)
10294, 100, 101syl2anc 692 1 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wex 1702  wcel 1988  wne 2791  wrex 2910  Vcvv 3195  cin 3566  wss 3567  c0 3907  𝒫 cpw 4149   ciun 4511   class class class wbr 4644  cmpt 4720   × cxp 5102  dom cdm 5104  ran crn 5105   Fn wfn 5871  wf 5872  ontowfo 5874  1-1-ontowf1o 5875  (class class class)co 6635  ωcom 7050  𝑚 cmap 7842  cen 7937  cdom 7938  Fincfn 7940  cardccrd 8746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-seqom 7528  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-oi 8400  df-card 8750  df-acn 8753
This theorem is referenced by:  inffien  8871  isnumbasgrplem3  37494
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