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Theorem fornex 7651
Description: If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
Assertion
Ref Expression
fornex (𝐴𝐶 → (𝐹:𝐴onto𝐵𝐵 ∈ V))

Proof of Theorem fornex
StepHypRef Expression
1 fofun 6587 . . . 4 (𝐹:𝐴onto𝐵 → Fun 𝐹)
2 funrnex 7649 . . . 4 (dom 𝐹𝐶 → (Fun 𝐹 → ran 𝐹 ∈ V))
31, 2syl5com 31 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶 → ran 𝐹 ∈ V))
4 fof 6586 . . . . 5 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
54fdmd 6519 . . . 4 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
65eleq1d 2901 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶𝐴𝐶))
7 forn 6589 . . . 4 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
87eleq1d 2901 . . 3 (𝐹:𝐴onto𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V))
93, 6, 83imtr3d 294 . 2 (𝐹:𝐴onto𝐵 → (𝐴𝐶𝐵 ∈ V))
109com12 32 1 (𝐴𝐶 → (𝐹:𝐴onto𝐵𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3499  dom cdm 5553  ran crn 5554  Fun wfun 6345  ontowfo 6349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359
This theorem is referenced by:  f1dmex  7652  f1ovv  7653  f1oeng  8520  fodomnum  9475  ttukeylem1  9923  fodomb  9940  cnexALT  12378  imasbas  16777  imasds  16778  elqtop  22223  qtoprest  22243  indishmph  22324  imasf1oxmet  22902  foresf1o  30181  noprc  33135  sge0f1o  42532  sge0fodjrnlem  42566
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