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Theorem f1oeng 8240
Description: The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)
Assertion
Ref Expression
f1oeng ((𝐴𝐶𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)

Proof of Theorem f1oeng
StepHypRef Expression
1 f1ofo 6384 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
2 fornex 7396 . . . 4 (𝐴𝐶 → (𝐹:𝐴onto𝐵𝐵 ∈ V))
31, 2syl5 34 . . 3 (𝐴𝐶 → (𝐹:𝐴1-1-onto𝐵𝐵 ∈ V))
43imp 397 . 2 ((𝐴𝐶𝐹:𝐴1-1-onto𝐵) → 𝐵 ∈ V)
5 f1oen2g 8238 . . 3 ((𝐴𝐶𝐵 ∈ V ∧ 𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
653com23 1162 . 2 ((𝐴𝐶𝐹:𝐴1-1-onto𝐵𝐵 ∈ V) → 𝐴𝐵)
74, 6mpd3an3 1592 1 ((𝐴𝐶𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wcel 2166  Vcvv 3413   class class class wbr 4872  ontowfo 6120  1-1-ontowf1o 6121  cen 8218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-reu 3123  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-en 8222
This theorem is referenced by:  f1oen  8242  f1imaeng  8281  f1dmvrnfibi  8518  onacda  9333  fictb  9381  canthp1lem2  9789  unbenlem  15982  4sqlem11  16029  conjsubgen  18043  dis2ndc  21633  ovoliunlem1  23667  logfac2  25354  rabfodom  29891  eulerpartlemgs2  30986  matunitlindflem2  33949
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