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Theorem relrnfvex 5156
Description: If a function has a set range, then the function value exists unconditional on the domain. (Contributed by Mario Carneiro, 24-May-2019.)
Assertion
Ref Expression
relrnfvex ((Rel 𝐹 ∧ ran 𝐹 ∈ V) → (𝐹𝐴) ∈ V)

Proof of Theorem relrnfvex
StepHypRef Expression
1 relfvssunirn 5154 . 2 (Rel 𝐹 → (𝐹𝐴) ⊆ ran 𝐹)
2 uniexg 4147 . 2 (ran 𝐹 ∈ V → ran 𝐹 ∈ V)
3 ssexg 3893 . 2 (((𝐹𝐴) ⊆ ran 𝐹 ran 𝐹 ∈ V) → (𝐹𝐴) ∈ V)
41, 2, 3syl2an 273 1 ((Rel 𝐹 ∧ ran 𝐹 ∈ V) → (𝐹𝐴) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wcel 1393  Vcvv 2554  wss 2914   cuni 3577  ran crn 4309  Rel wrel 4313  cfv 4865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941  ax-un 4142
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-br 3762  df-opab 3816  df-xp 4314  df-rel 4315  df-cnv 4316  df-dm 4318  df-rn 4319  df-iota 4830  df-fv 4873
This theorem is referenced by: (None)
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