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Theorem fornex 6064
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 5394 . . . 4 (𝐹:𝐴onto𝐵 → Fun 𝐹)
2 funrnex 6063 . . . 4 (dom 𝐹𝐶 → (Fun 𝐹 → ran 𝐹 ∈ V))
31, 2syl5com 29 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶 → ran 𝐹 ∈ V))
4 fof 5393 . . . . 5 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
5 fdm 5326 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
64, 5syl 14 . . . 4 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
76eleq1d 2226 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶𝐴𝐶))
8 forn 5396 . . . 4 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
98eleq1d 2226 . . 3 (𝐹:𝐴onto𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V))
103, 7, 93imtr3d 201 . 2 (𝐹:𝐴onto𝐵 → (𝐴𝐶𝐵 ∈ V))
1110com12 30 1 (𝐴𝐶 → (𝐹:𝐴onto𝐵𝐵 ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1335  wcel 2128  Vcvv 2712  dom cdm 4587  ran crn 4588  Fun wfun 5165  wf 5167  ontowfo 5169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4080  ax-sep 4083  ax-pow 4136  ax-pr 4170  ax-un 4394
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179
This theorem is referenced by:  f1dmex  6065  f1oeng  6703  ctfoex  7063  omctfn  12214
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