MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fornex Structured version   Visualization version   GIF version

Theorem fornex 7177
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 6154 . . . 4 (𝐹:𝐴onto𝐵 → Fun 𝐹)
2 funrnex 7175 . . . 4 (dom 𝐹𝐶 → (Fun 𝐹 → ran 𝐹 ∈ V))
31, 2syl5com 31 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶 → ran 𝐹 ∈ V))
4 fof 6153 . . . . 5 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
5 fdm 6089 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
64, 5syl 17 . . . 4 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
76eleq1d 2715 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶𝐴𝐶))
8 forn 6156 . . . 4 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
98eleq1d 2715 . . 3 (𝐹:𝐴onto𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V))
103, 7, 93imtr3d 282 . 2 (𝐹:𝐴onto𝐵 → (𝐴𝐶𝐵 ∈ V))
1110com12 32 1 (𝐴𝐶 → (𝐹:𝐴onto𝐵𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  dom cdm 5143  ran crn 5144  Fun wfun 5920  wf 5922  ontowfo 5924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934
This theorem is referenced by:  f1dmex  7178  f1ovv  7179  f1oeng  8016  fodomnum  8918  ttukeylem1  9369  fodomb  9386  cnexALT  11866  imasbas  16219  imasds  16220  elqtop  21548  qtoprest  21568  indishmph  21649  imasf1oxmet  22227  foresf1o  29469  noprc  32020  sge0f1o  40917  sge0fodjrnlem  40951
  Copyright terms: Public domain W3C validator