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Theorem f1oen2g 6642
Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6644 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
f1oen2g ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)

Proof of Theorem f1oen2g
StepHypRef Expression
1 f1of 5360 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
2 fex2 5286 . . . 4 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
31, 2syl3an1 1249 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
433coml 1188 . 2 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1-onto𝐵) → 𝐹 ∈ V)
5 simp3 983 . 2 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1-onto𝐵) → 𝐹:𝐴1-1-onto𝐵)
6 f1oen3g 6641 . 2 ((𝐹 ∈ V ∧ 𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
74, 5, 6syl2anc 408 1 ((𝐴𝑉𝐵𝑊𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 962  wcel 1480  Vcvv 2681   class class class wbr 3924  wf 5114  1-1-ontowf1o 5117  cen 6625
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-en 6628
This theorem is referenced by:  f1oeng  6644  enrefg  6651  en2d  6655  en3d  6656  ener  6666  f1imaen2g  6680  cnven  6695  xpcomen  6714  xpfi  6811  nnenom  10200
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