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Theorem f1oen2g 8989
Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 8992 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
f1oen2g ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)

Proof of Theorem f1oen2g
StepHypRef Expression
1 f1of 6838 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
2 fex2 7942 . . . 4 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
31, 2syl3an1 1160 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
433coml 1124 . 2 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1-onto𝐵) → 𝐹 ∈ V)
5 simp3 1135 . 2 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1-onto𝐵) → 𝐹:𝐴1-1-onto𝐵)
6 f1oen3g 8987 . 2 ((𝐹 ∈ V ∧ 𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
74, 5, 6syl2anc 582 1 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084  wcel 2098  Vcvv 3461   class class class wbr 5149  wf 6545  1-1-ontowf1o 6548  cen 8961
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-en 8965
This theorem is referenced by:  f1oeng  8992  enrefg  9005  en2d  9009  en3d  9010  ener  9022  f1imaen2g  9036  cnven  9059  xpcomen  9091  omxpen  9102  pw2eng  9106  unfilem3  9340  xpfiOLD  9347  hsmexlem1  10456  iccen  13514  uzenom  13970  nnenom  13986  eqgen  19149  dfod2  19536  hmphen  23738  clwlkclwwlken  29899  clwwlken  29939  clwwlknonclwlknonen  30250  dlwwlknondlwlknonen  30253
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