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Theorem fornex 7646
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 6584 . . . 4 (𝐹:𝐴onto𝐵 → Fun 𝐹)
2 funrnex 7644 . . . 4 (dom 𝐹𝐶 → (Fun 𝐹 → ran 𝐹 ∈ V))
31, 2syl5com 31 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶 → ran 𝐹 ∈ V))
4 fof 6583 . . . . 5 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
54fdmd 6516 . . . 4 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
65eleq1d 2894 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶𝐴𝐶))
7 forn 6586 . . . 4 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
87eleq1d 2894 . . 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 2105  Vcvv 3492  dom cdm 5548  ran crn 5549  Fun wfun 6342  ontowfo 6346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356
This theorem is referenced by:  f1dmex  7647  f1ovv  7648  f1oeng  8516  fodomnum  9471  ttukeylem1  9919  fodomb  9936  cnexALT  12373  imasbas  16773  imasds  16774  elqtop  22233  qtoprest  22253  indishmph  22334  imasf1oxmet  22912  foresf1o  30192  noprc  33146  sge0f1o  42541  sge0fodjrnlem  42575
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