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Theorem f1oen3g 8526
 Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 8529 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 6587 . . . 4 (𝑓 = 𝐹 → (𝑓:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵))
21spcegv 3546 . . 3 (𝐹𝑉 → (𝐹:𝐴1-1-onto𝐵 → ∃𝑓 𝑓:𝐴1-1-onto𝐵))
32imp 410 . 2 ((𝐹𝑉𝐹:𝐴1-1-onto𝐵) → ∃𝑓 𝑓:𝐴1-1-onto𝐵)
4 bren 8519 . 2 (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
53, 4sylibr 237 1 ((𝐹𝑉𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1781   ∈ wcel 2111   class class class wbr 5034  –1-1-onto→wf1o 6331   ≈ cen 8507 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-en 8511 This theorem is referenced by:  f1oen2g  8527  unen  8597  domdifsn  8601  domunsncan  8618  sucdom2  8628  sbthlem10  8638  domssex  8680  phplem2  8699  pssnn  8738  f1finf1o  8747  oien  9004  infdifsn  9122  fin4en1  9738  fin23lem21  9768  hashf1lem2  13830  odinf  18703  gsumval3lem2  19040  gsumval3  19041  hmphen2  22445  fnpreimac  30478  pibt2  34985
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