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Theorem relrnfvex 5221
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 5219 . 2 (Rel 𝐹 → (𝐹𝐴) ⊆ ran 𝐹)
2 uniexg 4203 . 2 (ran 𝐹 ∈ V → ran 𝐹 ∈ V)
3 ssexg 3924 . 2 (((𝐹𝐴) ⊆ ran 𝐹 ran 𝐹 ∈ V) → (𝐹𝐴) ∈ V)
41, 2, 3syl2an 277 1 ((Rel 𝐹 ∧ ran 𝐹 ∈ V) → (𝐹𝐴) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wcel 1409  Vcvv 2574  wss 2945   cuni 3608  ran crn 4374  Rel wrel 4378  cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381  df-dm 4383  df-rn 4384  df-iota 4895  df-fv 4938
This theorem is referenced by: (None)
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