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Theorem focdmex 7979
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 6822 . . . 4 (𝐹:𝐴onto𝐵 → Fun 𝐹)
2 funrnex 7977 . . . 4 (dom 𝐹𝐶 → (Fun 𝐹 → ran 𝐹 ∈ V))
31, 2syl5com 31 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶 → ran 𝐹 ∈ V))
4 fof 6821 . . . . 5 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
54fdmd 6747 . . . 4 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
65eleq1d 2824 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶𝐴𝐶))
7 forn 6824 . . . 4 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
87eleq1d 2824 . . 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 2106  Vcvv 3478  dom cdm 5689  ran crn 5690  Fun wfun 6557  ontowfo 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571
This theorem is referenced by:  f1dmex  7980  f1ovv  7981  fsetprcnex  8901  f1oeng  9010  fodomnum  10095  ttukeylem1  10547  fodomb  10564  cnexALT  13026  hasheqf1oi  14387  imasbas  17559  imasds  17560  elqtop  23721  qtoprest  23741  indishmph  23822  imasf1oxmet  24401  noprc  27839  foresf1o  32532  sge0f1o  46338  sge0fodjrnlem  46372
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