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Theorem focdmex 7996
Description: If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
Assertion
Ref Expression
focdmex (𝐴𝐶 → (𝐹:𝐴onto𝐵𝐵 ∈ V))

Proof of Theorem focdmex
StepHypRef Expression
1 fofun 6835 . . . 4 (𝐹:𝐴onto𝐵 → Fun 𝐹)
2 funrnex 7994 . . . 4 (dom 𝐹𝐶 → (Fun 𝐹 → ran 𝐹 ∈ V))
31, 2syl5com 31 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶 → ran 𝐹 ∈ V))
4 fof 6834 . . . . 5 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
54fdmd 6757 . . . 4 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
65eleq1d 2829 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶𝐴𝐶))
7 forn 6837 . . . 4 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
87eleq1d 2829 . . 3 (𝐹:𝐴onto𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V))
93, 6, 83imtr3d 293 . 2 (𝐹:𝐴onto𝐵 → (𝐴𝐶𝐵 ∈ V))
109com12 32 1 (𝐴𝐶 → (𝐹:𝐴onto𝐵𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  Vcvv 3488  dom cdm 5700  ran crn 5701  Fun wfun 6567  ontowfo 6571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581
This theorem is referenced by:  f1dmex  7997  f1ovv  7998  fsetprcnex  8920  f1oeng  9031  fodomnum  10126  ttukeylem1  10578  fodomb  10595  cnexALT  13051  hasheqf1oi  14400  imasbas  17572  imasds  17573  elqtop  23726  qtoprest  23746  indishmph  23827  imasf1oxmet  24406  noprc  27842  foresf1o  32532  sge0f1o  46303  sge0fodjrnlem  46337
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