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Theorem focdmex 6135
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 5455 . . . 4 (𝐹:𝐴onto𝐵 → Fun 𝐹)
2 funrnex 6134 . . . 4 (dom 𝐹𝐶 → (Fun 𝐹 → ran 𝐹 ∈ V))
31, 2syl5com 29 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶 → ran 𝐹 ∈ V))
4 fof 5454 . . . . 5 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
5 fdm 5387 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
64, 5syl 14 . . . 4 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
76eleq1d 2258 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶𝐴𝐶))
8 forn 5457 . . . 4 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
98eleq1d 2258 . . 3 (𝐹:𝐴onto𝐵 → (ran 𝐹 ∈ V ↔ 𝐵 ∈ V))
103, 7, 93imtr3d 202 . 2 (𝐹:𝐴onto𝐵 → (𝐴𝐶𝐵 ∈ V))
1110com12 30 1 (𝐴𝐶 → (𝐹:𝐴onto𝐵𝐵 ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  Vcvv 2752  dom cdm 4641  ran crn 4642  Fun wfun 5226  wf 5228  ontowfo 5230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240
This theorem is referenced by:  f1dmex  6136  f1oeng  6778  ctfoex  7142  ennnfonelemj0  12447  ennnfonelemg  12449  omctfn  12489  imasival  12776  imasbas  12777  imasplusg  12778
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