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Theorem f1oen3g 8909
Description: The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 8914 does not require the Axiom of Replacement nor the Axiom of Power Sets. (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 6773 . . . 4 (𝑓 = 𝐹 → (𝑓:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵))
21spcegv 3555 . . 3 (𝐹𝑉 → (𝐹:𝐴1-1-onto𝐵 → ∃𝑓 𝑓:𝐴1-1-onto𝐵))
32imp 408 . 2 ((𝐹𝑉𝐹:𝐴1-1-onto𝐵) → ∃𝑓 𝑓:𝐴1-1-onto𝐵)
4 bren 8896 . 2 (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
53, 4sylibr 233 1 ((𝐹𝑉𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wex 1782  wcel 2107   class class class wbr 5106  1-1-ontowf1o 6496  cen 8883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-en 8887
This theorem is referenced by:  f1oen2g  8911  unen  8993  domdifsn  9001  domunsncan  9019  sucdom2OLD  9029  sbthlem10  9039  domssex  9085  dif1enlemOLD  9104  pssnn  9115  f1oenfi  9129  f1oenfirn  9130  sbthfilem  9148  sucdom2  9153  phplem2OLD  9165  pssnnOLD  9212  f1finf1oOLD  9219  oien  9479  infdifsn  9598  fin4en1  10250  fin23lem21  10280  hashf1lem2  14361  odinf  19350  gsumval3lem2  19688  gsumval3  19689  hmphen2  23166  fnpreimac  31633  pibt2  35934
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