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Theorem f1oen3g 6728
Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6731 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 5429 . . . 4 (𝑓 = 𝐹 → (𝑓:𝐴1-1-onto𝐵𝐹:𝐴1-1-onto𝐵))
21spcegv 2818 . . 3 (𝐹𝑉 → (𝐹:𝐴1-1-onto𝐵 → ∃𝑓 𝑓:𝐴1-1-onto𝐵))
32imp 123 . 2 ((𝐹𝑉𝐹:𝐴1-1-onto𝐵) → ∃𝑓 𝑓:𝐴1-1-onto𝐵)
4 bren 6721 . 2 (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
53, 4sylibr 133 1 ((𝐹𝑉𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wex 1485  wcel 2141   class class class wbr 3987  1-1-ontowf1o 5195  cen 6712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-en 6715
This theorem is referenced by:  f1oen2g  6729  unen  6790  phplem2  6827  sbthlemi10  6939
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