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Theorem infpwfien 8837
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 8792 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
2 infn0 8174 . . . . . . . 8 (ω ≼ 𝐴𝐴 ≠ ∅)
32adantl 482 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≠ ∅)
4 fseqen 8802 . . . . . . 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 8009 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (ω × 𝐴) ≼ (𝐴 × 𝐴))
7 domentr 7967 . . . . . . 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 7966 . . . . . 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 8813 . . . . 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 4495 . . . . . . . . 9 (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↔ ∃𝑛 ∈ ω 𝑥 ∈ (𝐴𝑚 𝑛))
14 elmapi 7831 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝐴𝑚 𝑛) → 𝑥:𝑛𝐴)
1514ad2antll 764 . . . . . . . . . . . . . 14 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴𝑚 𝑛))) → 𝑥:𝑛𝐴)
16 frn 6015 . . . . . . . . . . . . . 14 (𝑥:𝑛𝐴 → ran 𝑥𝐴)
1715, 16syl 17 . . . . . . . . . . . . 13 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴𝑚 𝑛))) → ran 𝑥𝐴)
18 vex 3192 . . . . . . . . . . . . . . 15 𝑥 ∈ V
1918rnex 7054 . . . . . . . . . . . . . 14 ran 𝑥 ∈ V
2019elpw 4141 . . . . . . . . . . . . 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 3608 . . . . . . . . . . . . . 14 𝑛𝑛
24 ssnnfi 8131 . . . . . . . . . . . . . 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 6007 . . . . . . . . . . . . . . 15 (𝑥:𝑛𝐴𝑥 Fn 𝑛)
27 dffn4 6083 . . . . . . . . . . . . . . 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 8204 . . . . . . . . . . . . 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 3781 . . . . . . . . . . 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 3025 . . . . . . . . 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 2621 . . . . . . 7 (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥) = (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥)
3836, 37fmptd 6346 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥): 𝑛 ∈ ω (𝐴𝑚 𝑛)⟶(𝒫 𝐴 ∩ Fin))
39 ffn 6007 . . . . . 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 6015 . . . . . . 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 3817 . . . . . . . . . . . 12 (𝒫 𝐴 ∩ Fin) ⊆ Fin
44 simpr 477 . . . . . . . . . . . 12 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
4543, 44sseldi 3585 . . . . . . . . . . 11 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ Fin)
46 isfi 7931 . . . . . . . . . . 11 (𝑦 ∈ Fin ↔ ∃𝑚 ∈ ω 𝑦𝑚)
4745, 46sylib 208 . . . . . . . . . 10 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑚 ∈ ω 𝑦𝑚)
48 ensym 7957 . . . . . . . . . . . . 13 (𝑦𝑚𝑚𝑦)
49 bren 7916 . . . . . . . . . . . . 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 6099 . . . . . . . . . . . . . . . . . . . 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 3816 . . . . . . . . . . . . . . . . . . . . 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 3585 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑦 ∈ 𝒫 𝐴)
5756elpwid 4146 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑦𝐴)
5853, 57fssd 6019 . . . . . . . . . . . . . . . . . 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 3192 . . . . . . . . . . . . . . . . . . 19 𝑚 ∈ V
61 elmapg 7822 . . . . . . . . . . . . . . . . . . 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 6618 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑚 → (𝐴𝑚 𝑛) = (𝐴𝑚 𝑚))
6564eleq2d 2684 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (𝑥 ∈ (𝐴𝑚 𝑛) ↔ 𝑥 ∈ (𝐴𝑚 𝑚)))
6665rspcev 3298 . . . . . . . . . . . . . . . . 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 6106 . . . . . . . . . . . . . . . . . 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 6080 . . . . . . . . . . . . . . . . 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 2627 . . . . . . . . . . . . . . 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 1843 . . . . . . . . . . . 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 3025 . . . . . . . . . 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 3192 . . . . . . . . . 10 𝑦 ∈ V
8237elrnmpt 5337 . . . . . . . . . 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 2913 . . . . . . . . 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 3593 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ⊆ ran (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥))
8842, 87eqssd 3604 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ran (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥) = (𝒫 𝐴 ∩ Fin))
89 df-fo 5858 . . . . 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 8832 . . . 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 7961 . . 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 4815 . . . . 5 (𝐴 ∈ dom card → 𝒫 𝐴 ∈ V)
9695adantr 481 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝒫 𝐴 ∈ V)
97 inex1g 4766 . . . 4 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
9896, 97syl 17 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ∈ V)
99 infpwfidom 8803 . . 3 ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
10098, 99syl 17 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
101 sbth 8032 . 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 1480  wex 1701  wcel 1987  wne 2790  wrex 2908  Vcvv 3189  cin 3558  wss 3559  c0 3896  𝒫 cpw 4135   ciun 4490   class class class wbr 4618  cmpt 4678   × cxp 5077  dom cdm 5079  ran crn 5080   Fn wfn 5847  wf 5848  ontowfo 5850  1-1-ontowf1o 5851  (class class class)co 6610  ωcom 7019  𝑚 cmap 7809  cen 7904  cdom 7905  Fincfn 7907  cardccrd 8713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-seqom 7495  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-oi 8367  df-card 8717  df-acn 8720
This theorem is referenced by:  inffien  8838  isnumbasgrplem3  37191
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