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Theorem relrnfvex 5267
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 5265 . 2 (Rel 𝐹 → (𝐹𝐴) ⊆ ran 𝐹)
2 uniexg 4228 . 2 (ran 𝐹 ∈ V → ran 𝐹 ∈ V)
3 ssexg 3943 . 2 (((𝐹𝐴) ⊆ ran 𝐹 ran 𝐹 ∈ V) → (𝐹𝐴) ∈ V)
41, 2, 3syl2an 283 1 ((Rel 𝐹 ∧ ran 𝐹 ∈ V) → (𝐹𝐴) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wcel 1434  Vcvv 2612  wss 2984   cuni 3627  ran crn 4401  Rel wrel 4405  cfv 4968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999  ax-un 4223
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-xp 4406  df-rel 4407  df-cnv 4408  df-dm 4410  df-rn 4411  df-iota 4933  df-fv 4976
This theorem is referenced by: (None)
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