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Theorem isfin1-4 9801
 Description: A set is I-finite iff every system of subsets contains a minimal subset. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
isfin1-4 (𝐴𝑉 → (𝐴 ∈ Fin ↔ [] Fr 𝒫 𝐴))

Proof of Theorem isfin1-4
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 isfin1-3 9800 . 2 (𝐴𝑉 → (𝐴 ∈ Fin ↔ [] Fr 𝒫 𝐴))
2 eqid 2825 . . . 4 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
32compssiso 9788 . . 3 (𝐴𝑉 → (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) Isom [] , [] (𝒫 𝐴, 𝒫 𝐴))
4 isofr 7090 . . 3 ((𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) Isom [] , [] (𝒫 𝐴, 𝒫 𝐴) → ( [] Fr 𝒫 𝐴 [] Fr 𝒫 𝐴))
53, 4syl 17 . 2 (𝐴𝑉 → ( [] Fr 𝒫 𝐴 [] Fr 𝒫 𝐴))
61, 5bitr4d 283 1 (𝐴𝑉 → (𝐴 ∈ Fin ↔ [] Fr 𝒫 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∈ wcel 2107   ∖ cdif 3936  𝒫 cpw 4541   ↦ cmpt 5142   Fr wfr 5509  ◡ccnv 5552   Isom wiso 6352   [⊊] crpss 7441  Fincfn 8501 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 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454 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 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  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-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-ov 7154  df-oprab 7155  df-mpo 7156  df-rpss 7442  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505 This theorem is referenced by: (None)
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