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Theorem infpwfien 9084
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 9039 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
2 infn0 8377 . . . . . . . 8 (ω ≼ 𝐴𝐴 ≠ ∅)
32adantl 467 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≠ ∅)
4 fseqen 9049 . . . . . . 7 (((𝐴 × 𝐴) ≈ 𝐴𝐴 ≠ ∅) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≈ (ω × 𝐴))
51, 3, 4syl2anc 565 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≈ (ω × 𝐴))
6 xpdom1g 8212 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (ω × 𝐴) ≼ (𝐴 × 𝐴))
7 domentr 8167 . . . . . . 7 (((ω × 𝐴) ≼ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ≈ 𝐴) → (ω × 𝐴) ≼ 𝐴)
86, 1, 7syl2anc 565 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (ω × 𝐴) ≼ 𝐴)
9 endomtr 8166 . . . . . 6 (( 𝑛 ∈ ω (𝐴𝑚 𝑛) ≈ (ω × 𝐴) ∧ (ω × 𝐴) ≼ 𝐴) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≼ 𝐴)
105, 8, 9syl2anc 565 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ≼ 𝐴)
11 numdom 9060 . . . . 5 ((𝐴 ∈ dom card ∧ 𝑛 ∈ ω (𝐴𝑚 𝑛) ≼ 𝐴) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ∈ dom card)
1210, 11syldan 571 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝑛 ∈ ω (𝐴𝑚 𝑛) ∈ dom card)
13 eliun 4656 . . . . . . . . 9 (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↔ ∃𝑛 ∈ ω 𝑥 ∈ (𝐴𝑚 𝑛))
14 elmapi 8030 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝐴𝑚 𝑛) → 𝑥:𝑛𝐴)
1514ad2antll 700 . . . . . . . . . . . . . 14 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴𝑚 𝑛))) → 𝑥:𝑛𝐴)
16 frn 6193 . . . . . . . . . . . . . 14 (𝑥:𝑛𝐴 → ran 𝑥𝐴)
1715, 16syl 17 . . . . . . . . . . . . 13 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴𝑚 𝑛))) → ran 𝑥𝐴)
18 vex 3352 . . . . . . . . . . . . . . 15 𝑥 ∈ V
1918rnex 7246 . . . . . . . . . . . . . 14 ran 𝑥 ∈ V
2019elpw 4301 . . . . . . . . . . . . 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 746 . . . . . . . . . . . . . 14 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴𝑚 𝑛))) → 𝑛 ∈ ω)
23 ssid 3771 . . . . . . . . . . . . . 14 𝑛𝑛
24 ssnnfi 8334 . . . . . . . . . . . . . 14 ((𝑛 ∈ ω ∧ 𝑛𝑛) → 𝑛 ∈ Fin)
2522, 23, 24sylancl 566 . . . . . . . . . . . . 13 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴𝑚 𝑛))) → 𝑛 ∈ Fin)
26 ffn 6185 . . . . . . . . . . . . . . 15 (𝑥:𝑛𝐴𝑥 Fn 𝑛)
27 dffn4 6262 . . . . . . . . . . . . . . 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 8407 . . . . . . . . . . . . 13 ((𝑛 ∈ Fin ∧ 𝑥:𝑛onto→ran 𝑥) → ran 𝑥 ∈ Fin)
3125, 29, 30syl2anc 565 . . . . . . . . . . . 12 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴𝑚 𝑛))) → ran 𝑥 ∈ Fin)
3221, 31elind 3947 . . . . . . . . . . 11 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝑛 ∈ ω ∧ 𝑥 ∈ (𝐴𝑚 𝑛))) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin))
3332expr 444 . . . . . . . . . 10 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑛 ∈ ω) → (𝑥 ∈ (𝐴𝑚 𝑛) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin)))
3433rexlimdva 3178 . . . . . . . . 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 393 . . . . . . 7 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛)) → ran 𝑥 ∈ (𝒫 𝐴 ∩ Fin))
37 eqid 2770 . . . . . . 7 (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥) = (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥)
3836, 37fmptd 6527 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥): 𝑛 ∈ ω (𝐴𝑚 𝑛)⟶(𝒫 𝐴 ∩ Fin))
39 ffn 6185 . . . . . 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 6193 . . . . . . 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 3980 . . . . . . . . . . . 12 (𝒫 𝐴 ∩ Fin) ⊆ Fin
44 simpr 471 . . . . . . . . . . . 12 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
4543, 44sseldi 3748 . . . . . . . . . . 11 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ Fin)
46 isfi 8132 . . . . . . . . . . 11 (𝑦 ∈ Fin ↔ ∃𝑚 ∈ ω 𝑦𝑚)
4745, 46sylib 208 . . . . . . . . . 10 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ∃𝑚 ∈ ω 𝑦𝑚)
48 ensym 8157 . . . . . . . . . . . . 13 (𝑦𝑚𝑚𝑦)
49 bren 8117 . . . . . . . . . . . . 13 (𝑚𝑦 ↔ ∃𝑥 𝑥:𝑚1-1-onto𝑦)
5048, 49sylib 208 . . . . . . . . . . . 12 (𝑦𝑚 → ∃𝑥 𝑥:𝑚1-1-onto𝑦)
51 simprl 746 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑚 ∈ ω)
52 f1of 6278 . . . . . . . . . . . . . . . . . . . 20 (𝑥:𝑚1-1-onto𝑦𝑥:𝑚𝑦)
5352ad2antll 700 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑥:𝑚𝑦)
54 inss1 3979 . . . . . . . . . . . . . . . . . . . . 21 (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
55 simplr 744 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
5654, 55sseldi 3748 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑦 ∈ 𝒫 𝐴)
5756elpwid 4307 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑦𝐴)
5853, 57fssd 6197 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑥:𝑚𝐴)
59 simplll 750 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝐴 ∈ dom card)
60 vex 3352 . . . . . . . . . . . . . . . . . . 19 𝑚 ∈ V
61 elmapg 8021 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ dom card ∧ 𝑚 ∈ V) → (𝑥 ∈ (𝐴𝑚 𝑚) ↔ 𝑥:𝑚𝐴))
6259, 60, 61sylancl 566 . . . . . . . . . . . . . . . . . 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 6800 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑚 → (𝐴𝑚 𝑛) = (𝐴𝑚 𝑚))
6564eleq2d 2835 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (𝑥 ∈ (𝐴𝑚 𝑛) ↔ 𝑥 ∈ (𝐴𝑚 𝑚)))
6665rspcev 3458 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ ω ∧ 𝑥 ∈ (𝐴𝑚 𝑚)) → ∃𝑛 ∈ ω 𝑥 ∈ (𝐴𝑚 𝑛))
6751, 63, 66syl2anc 565 . . . . . . . . . . . . . . . 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 6285 . . . . . . . . . . . . . . . . . 18 (𝑥:𝑚1-1-onto𝑦𝑥:𝑚onto𝑦)
7069ad2antll 700 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑥:𝑚onto𝑦)
71 forn 6259 . . . . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ (𝑚 ∈ ω ∧ 𝑥:𝑚1-1-onto𝑦)) → 𝑦 = ran 𝑥)
7468, 73jca 495 . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . 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 1997 . . . . . . . . . . . 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 3178 . . . . . . . . . 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 397 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → ∃𝑥(𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ 𝑦 = ran 𝑥)))
81 vex 3352 . . . . . . . . . 10 𝑦 ∈ V
8237elrnmpt 5510 . . . . . . . . . 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 3066 . . . . . . . . 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 3756 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ⊆ ran (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥))
8842, 87eqssd 3767 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ran (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥) = (𝒫 𝐴 ∩ Fin))
89 df-fo 6037 . . . . 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 564 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 𝑛 ∈ ω (𝐴𝑚 𝑛) ↦ ran 𝑥): 𝑛 ∈ ω (𝐴𝑚 𝑛)–onto→(𝒫 𝐴 ∩ Fin))
91 fodomnum 9079 . . . 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 8161 . . 3 (((𝒫 𝐴 ∩ Fin) ≼ 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ 𝑛 ∈ ω (𝐴𝑚 𝑛) ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≼ 𝐴)
9492, 10, 93syl2anc 565 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≼ 𝐴)
95 pwexg 4978 . . . . 5 (𝐴 ∈ dom card → 𝒫 𝐴 ∈ V)
9695adantr 466 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝒫 𝐴 ∈ V)
97 inex1g 4932 . . . 4 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
9896, 97syl 17 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ∈ V)
99 infpwfidom 9050 . . 3 ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
10098, 99syl 17 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
101 sbth 8235 . 2 (((𝒫 𝐴 ∩ Fin) ≼ 𝐴𝐴 ≼ (𝒫 𝐴 ∩ Fin)) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴)
10294, 100, 101syl2anc 565 1 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wex 1851  wcel 2144  wne 2942  wrex 3061  Vcvv 3349  cin 3720  wss 3721  c0 4061  𝒫 cpw 4295   ciun 4652   class class class wbr 4784  cmpt 4861   × cxp 5247  dom cdm 5249  ran crn 5250   Fn wfn 6026  wf 6027  ontowfo 6029  1-1-ontowf1o 6030  (class class class)co 6792  ωcom 7211  𝑚 cmap 8008  cen 8105  cdom 8106  Fincfn 8108  cardccrd 8960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-seqom 7695  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-oi 8570  df-card 8964  df-acn 8967
This theorem is referenced by:  inffien  9085  isnumbasgrplem3  38194
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