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 e. _V ) -> ( F ` A ) e. _V )

Proof of Theorem relrnfvex

Step	Hyp	Ref	Expression
1		relfvssunirn 5191	. 2 |- ( Rel F -> ( F ` A ) C_ U. ran F )
2		uniexg 4175	. 2 |- ( ran F e. _V -> U. ran F e. _V )
3		ssexg 3896	. 2 |- ( ( ( F ` A ) C_ U. ran F /\ U. ran F e. _V ) -> ( F ` A ) e. _V )
4	1, 2, 3	syl2an 273	1 |- ( ( Rel F /\ ran F e. _V ) -> ( F ` A ) e. _V )

Syntax hints:   -> wi 4   /\ wa 97   e. wcel 1393   _Vcvv 2557   C_ wss 2917   U.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)