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Theorem focdmex 7893
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 6762 . . . 4 (𝐹:𝐴onto𝐵 → Fun 𝐹)
2 funrnex 7891 . . . 4 (dom 𝐹𝐶 → (Fun 𝐹 → ran 𝐹 ∈ V))
31, 2syl5com 31 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶 → ran 𝐹 ∈ V))
4 fof 6761 . . . . 5 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
54fdmd 6684 . . . 4 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
65eleq1d 2823 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶𝐴𝐶))
7 forn 6764 . . . 4 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
87eleq1d 2823 . . 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 2107  Vcvv 3448  dom cdm 5638  ran crn 5639  Fun wfun 6495  ontowfo 6499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509
This theorem is referenced by:  f1dmex  7894  f1ovv  7895  fsetprcnex  8807  f1oeng  8918  fodomnum  10000  ttukeylem1  10452  fodomb  10469  cnexALT  12918  hasheqf1oi  14258  imasbas  17401  imasds  17402  elqtop  23064  qtoprest  23084  indishmph  23165  imasf1oxmet  23744  noprc  27141  foresf1o  31473  sge0f1o  44697  sge0fodjrnlem  44731
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