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Theorem inffien 8769
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 8768 . . . . . . . 8 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴)
2 relen 7846 . . . . . . . . 9 Rel ≈
32brrelexi 5082 . . . . . . . 8 ((𝒫 𝐴 ∩ Fin) ≈ 𝐴 → (𝒫 𝐴 ∩ Fin) ∈ V)
41, 3syl 17 . . . . . . 7 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ∈ V)
5 difss 3699 . . . . . . 7 ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ⊆ (𝒫 𝐴 ∩ Fin)
6 ssdomg 7887 . . . . . . 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 7901 . . . . . 6 ((((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ (𝒫 𝐴 ∩ Fin) ∧ (𝒫 𝐴 ∩ Fin) ≈ 𝐴) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ 𝐴)
97, 1, 8syl2anc 691 . . . . 5 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ 𝐴)
10 numdom 8744 . . . . 5 ((𝐴 ∈ dom card ∧ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ 𝐴) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∈ dom card)
119, 10syldan 486 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∈ dom card)
12 eqid 2610 . . . . . 6 (𝑥 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ 𝑥) = (𝑥 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ 𝑥)
1312fifo 8221 . . . . 5 (𝐴 ∈ dom card → (𝑥 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ 𝑥):((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
1413adantr 480 . . . 4 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ 𝑥):((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
15 fodomnum 8763 . . . 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 63 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (fi‘𝐴) ≼ ((𝒫 𝐴 ∩ Fin) ∖ {∅}))
17 domtr 7895 . . 3 (((fi‘𝐴) ≼ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ 𝐴) → (fi‘𝐴) ≼ 𝐴)
1816, 9, 17syl2anc 691 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (fi‘𝐴) ≼ 𝐴)
19 fvex 6113 . . 3 (fi‘𝐴) ∈ V
20 ssfii 8208 . . . 4 (𝐴 ∈ dom card → 𝐴 ⊆ (fi‘𝐴))
2120adantr 480 . . 3 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ⊆ (fi‘𝐴))
22 ssdomg 7887 . . 3 ((fi‘𝐴) ∈ V → (𝐴 ⊆ (fi‘𝐴) → 𝐴 ≼ (fi‘𝐴)))
2319, 21, 22mpsyl 66 . 2 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≼ (fi‘𝐴))
24 sbth 7965 . 2 (((fi‘𝐴) ≼ 𝐴𝐴 ≼ (fi‘𝐴)) → (fi‘𝐴) ≈ 𝐴)
2518, 23, 24syl2anc 691 1 ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (fi‘𝐴) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 1977  Vcvv 3173  cdif 3537  cin 3539  wss 3540  c0 3874  𝒫 cpw 4108  {csn 4125   cint 4410   class class class wbr 4583  cmpt 4643  dom cdm 5038  ontowfo 5802  cfv 5804  ωcom 6957  cen 7838  cdom 7839  Fincfn 7841  ficfi 8199  cardccrd 8644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-oi 8298  df-card 8648  df-acn 8651
This theorem is referenced by: (None)
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