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Theorem f1oen2g 7552
Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 7554 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
f1oen2g

Proof of Theorem f1oen2g
StepHypRef Expression
1 f1of 5821 . . . 4
2 fex2 6755 . . . 4
31, 2syl3an1 1261 . . 3
433coml 1203 . 2
5 simp3 998 . 2
6 f1oen3g 7551 . 2
74, 5, 6syl2anc 661 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\w3a 973  e.wcel 1818   cvv 3109   class class class wbr 4452  -->wf 5589  -1-1-onto->wf1o 5592   cen 7533
This theorem is referenced by:  f1oeng  7554  enrefg  7567  en2d  7571  en3d  7572  ener  7582  f1imaen2g  7596  cnven  7611  xpcomen  7628  omxpen  7639  pw2eng  7643  unfilem3  7806  xpfi  7811  hsmexlem1  8827  iccen  11694  uzenom  12075  nnenom  12090  eqgen  16254  dfod2  16586  hmphen  20286  0sgmppw  23473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-en 7537
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