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Theorem fornex 6094
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 5421 . . . 4 (𝐹:𝐴onto𝐵 → Fun 𝐹)
2 funrnex 6093 . . . 4 (dom 𝐹𝐶 → (Fun 𝐹 → ran 𝐹 ∈ V))
31, 2syl5com 29 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶 → ran 𝐹 ∈ V))
4 fof 5420 . . . . 5 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
5 fdm 5353 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
64, 5syl 14 . . . 4 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
76eleq1d 2239 . . 3 (𝐹:𝐴onto𝐵 → (dom 𝐹𝐶𝐴𝐶))
8 forn 5423 . . . 4 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
98eleq1d 2239 . . 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 1348  wcel 2141  Vcvv 2730  dom cdm 4611  ran crn 4612  Fun wfun 5192  wf 5194  ontowfo 5196
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206
This theorem is referenced by:  f1dmex  6095  f1oeng  6735  ctfoex  7095  omctfn  12398
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