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

Theorem fornex 7082
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 6073 . . . 4 (𝐹:𝐴onto𝐵 → Fun 𝐹)
2 funrnex 7080 . . . 4 (dom 𝐹𝐶 → (Fun 𝐹 → ran 𝐹 ∈ V))
31, 2syl5com 31 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶 → ran 𝐹 ∈ V))
4 fof 6072 . . . . 5 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
5 fdm 6008 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
64, 5syl 17 . . . 4 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
76eleq1d 2683 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶𝐴𝐶))
8 forn 6075 . . . 4 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
98eleq1d 2683 . . 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 1480  wcel 1987  Vcvv 3186  dom cdm 5074  ran crn 5075  Fun wfun 5841  wf 5843  ontowfo 5845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855
This theorem is referenced by:  f1dmex  7083  f1ovv  7084  f1oeng  7918  fodomnum  8824  ttukeylem1  9275  fodomb  9292  cnexALT  11772  imasbas  16093  imasds  16094  elqtop  21410  qtoprest  21430  indishmph  21511  imasf1oxmet  22090  foresf1o  29187  noprc  31541  sge0f1o  39903  sge0fodjrnlem  39937
  Copyright terms: Public domain W3C validator