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Theorem relrnfvex 5193
 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 5191 . 2 (Rel 𝐹 → (𝐹𝐴) ⊆ ran 𝐹)
2 uniexg 4175 . 2 (ran 𝐹 ∈ V → ran 𝐹 ∈ V)
3 ssexg 3896 . 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 2557   ⊆ wss 2917  ∪ cuni 3580  ran crn 4346  Rel wrel 4350  ‘cfv 4902 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 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 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 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-iota 4867  df-fv 4910 This theorem is referenced by: (None)
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