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Theorem f1oen3g 6170
 Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6173 does not require the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
f1oen3g ((𝐹 𝑉 𝐹:A1-1-ontoB) → AB)

Proof of Theorem f1oen3g
Dummy variable f is distinct from all other variables.
StepHypRef Expression
1 f1oeq1 5060 . . . 4 (f = 𝐹 → (f:A1-1-ontoB𝐹:A1-1-ontoB))
21spcegv 2635 . . 3 (𝐹 𝑉 → (𝐹:A1-1-ontoBf f:A1-1-ontoB))
32imp 115 . 2 ((𝐹 𝑉 𝐹:A1-1-ontoB) → f f:A1-1-ontoB)
4 bren 6164 . 2 (ABf f:A1-1-ontoB)
53, 4sylibr 137 1 ((𝐹 𝑉 𝐹:A1-1-ontoB) → AB)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1378   ∈ wcel 1390   class class class wbr 3755  –1-1-onto→wf1o 4844   ≈ cen 6155 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-en 6158 This theorem is referenced by:  f1oen2g  6171  unen  6229
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