Theorem relrnfvex 5552

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 5550	. 2 |- ( Rel F -> ( F ` A ) C_ U. ran F )
2		uniexg 4457	. 2 |- ( ran F e. _V -> U. ran F e. _V )
3		ssexg 4157	. 2 |- ( ( ( F ` A ) C_ U. ran F /\ U. ran F e. _V ) -> ( F ` A ) e. _V )
4	1, 2, 3	syl2an 289	1 |- ( ( Rel F /\ ran F e. _V ) -> ( F ` A ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2160   _Vcvv 2752   C_ wss 3144   U.cuni 3824   ran crn 4645   Rel wrel 4649   ` cfv 5235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-dm 4654  df-rn 4655  df-iota 5196  df-fv 5243

This theorem is referenced by: (None)