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Theorem f1oen2g 7924
Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 7926 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 6099 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
2 fex2 7075 . . . 4 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
31, 2syl3an1 1356 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
433coml 1269 . 2 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1-onto𝐵) → 𝐹 ∈ V)
5 simp3 1061 . 2 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1-onto𝐵) → 𝐹:𝐴1-1-onto𝐵)
6 f1oen3g 7923 . 2 ((𝐹 ∈ V ∧ 𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
74, 5, 6syl2anc 692 1 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036  wcel 1987  Vcvv 3189   class class class wbr 4618  wf 5848  1-1-ontowf1o 5851  cen 7904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-en 7908
This theorem is referenced by:  f1oeng  7926  enrefg  7939  en2d  7943  en3d  7944  ener  7954  enerOLD  7955  f1imaen2g  7969  cnven  7984  xpcomen  8003  omxpen  8014  pw2eng  8018  unfilem3  8178  xpfi  8183  hsmexlem1  9200  iccen  12267  uzenom  12711  nnenom  12727  eqgen  17579  dfod2  17913  hmphen  21511  0sgmppw  24840
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