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Theorem inffien 9481
Description: The set of finite intersections of an infinite well-orderable set is equinumerous to the set itself. (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
inffien ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (fi‘𝐴) ≈ 𝐴)

Proof of Theorem inffien
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 infpwfien 9480 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴)
2 relen 8506 . . . . . . . . 9 Rel ≈
32brrelex1i 5606 . . . . . . . 8 ((𝒫 𝐴 ∩ Fin) ≈ 𝐴 → (𝒫 𝐴 ∩ Fin) ∈ V)
41, 3syl 17 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ∈ V)
5 difss 4111 . . . . . . 7 ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ⊆ (𝒫 𝐴 ∩ Fin)
6 ssdomg 8547 . . . . . . 7 ((𝒫 𝐴 ∩ Fin) ∈ V → (((𝒫 𝐴 ∩ Fin) ∖ {∅}) ⊆ (𝒫 𝐴 ∩ Fin) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ (𝒫 𝐴 ∩ Fin)))
74, 5, 6mpisyl 21 . . . . . 6 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ (𝒫 𝐴 ∩ Fin))
8 domentr 8560 . . . . . 6 ((((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ (𝒫 𝐴 ∩ Fin) ∧ (𝒫 𝐴 ∩ Fin) ≈ 𝐴) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ 𝐴)
97, 1, 8syl2anc 584 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ 𝐴)
10 numdom 9456 . . . . 5 ((𝐴 ∈ dom card ∧ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ 𝐴) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∈ dom card)
119, 10syldan 591 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∈ dom card)
12 eqid 2824 . . . . . 6 (𝑥 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ 𝑥) = (𝑥 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ 𝑥)
1312fifo 8888 . . . . 5 (𝐴 ∈ dom card → (𝑥 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ 𝑥):((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
1413adantr 481 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ 𝑥):((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
15 fodomnum 9475 . . . 4 (((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∈ dom card → ((𝑥 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ 𝑥):((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴) → (fi‘𝐴) ≼ ((𝒫 𝐴 ∩ Fin) ∖ {∅})))
1611, 14, 15sylc 65 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (fi‘𝐴) ≼ ((𝒫 𝐴 ∩ Fin) ∖ {∅}))
17 domtr 8554 . . 3 (((fi‘𝐴) ≼ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ 𝐴) → (fi‘𝐴) ≼ 𝐴)
1816, 9, 17syl2anc 584 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (fi‘𝐴) ≼ 𝐴)
19 fvex 6679 . . 3 (fi‘𝐴) ∈ V
20 ssfii 8875 . . . 4 (𝐴 ∈ dom card → 𝐴 ⊆ (fi‘𝐴))
2120adantr 481 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ⊆ (fi‘𝐴))
22 ssdomg 8547 . . 3 ((fi‘𝐴) ∈ V → (𝐴 ⊆ (fi‘𝐴) → 𝐴 ≼ (fi‘𝐴)))
2319, 21, 22mpsyl 68 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≼ (fi‘𝐴))
24 sbth 8629 . 2 (((fi‘𝐴) ≼ 𝐴𝐴 ≼ (fi‘𝐴)) → (fi‘𝐴) ≈ 𝐴)
2518, 23, 24syl2anc 584 1 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (fi‘𝐴) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  Vcvv 3499  cdif 3936  cin 3938  wss 3939  c0 4294  𝒫 cpw 4541  {csn 4563   cint 4873   class class class wbr 5062  cmpt 5142  dom cdm 5553  ontowfo 6349  cfv 6351  ωcom 7571  cen 8498  cdom 8499  Fincfn 8501  ficfi 8866  cardccrd 9356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-seqom 8078  df-1o 8096  df-oadd 8100  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fi 8867  df-oi 8966  df-card 9360  df-acn 9363
This theorem is referenced by: (None)
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