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Theorem f1oen3g 7956
Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 7959 does not require the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
f1oen3g ((𝐹𝑉𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)

Proof of Theorem f1oen3g
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 f1oeq1 6114 . . . 4 (𝑓 = 𝐹 → (𝑓:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵))
21spcegv 3289 . . 3 (𝐹𝑉 → (𝐹:𝐴1-1-onto𝐵 → ∃𝑓 𝑓:𝐴1-1-onto𝐵))
32imp 445 . 2 ((𝐹𝑉𝐹:𝐴1-1-onto𝐵) → ∃𝑓 𝑓:𝐴1-1-onto𝐵)
4 bren 7949 . 2 (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
53, 4sylibr 224 1 ((𝐹𝑉𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wex 1702  wcel 1988   class class class wbr 4644  1-1-ontowf1o 5875  cen 7937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-en 7941
This theorem is referenced by:  f1oen2g  7957  unen  8025  domdifsn  8028  domunsncan  8045  sbthlem10  8064  domssex  8106  phplem2  8125  sucdom2  8141  pssnn  8163  f1finf1o  8172  oien  8428  infdifsn  8539  fin4en1  9116  fin23lem21  9146  hashf1lem2  13223  odinf  17961  gsumval3lem1  18287  gsumval3lem2  18288  gsumval3  18289  hmphen2  21583
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