Theorem relrnfvex 5503

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 5501	. 2 |- ( Rel F -> ( F ` A ) C_ U. ran F )
2		uniexg 4416	. 2 |- ( ran F e. _V -> U. ran F e. _V )
3		ssexg 4120	. 2 |- ( ( ( F ` A ) C_ U. ran F /\ U. ran F e. _V ) -> ( F ` A ) e. _V )
4	1, 2, 3	syl2an 287	1 |- ( ( Rel F /\ ran F e. _V ) -> ( F ` A ) e. _V )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   _Vcvv 2725   C_ wss 3115   U.cuni 3788   ran crn 4604   Rel wrel 4608   ` cfv 5187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-pow 4152  ax-pr 4186  ax-un 4410

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ral 2448  df-rex 2449  df-v 2727  df-un 3119  df-in 3121  df-ss 3128  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-br 3982  df-opab 4043  df-xp 4609  df-rel 4610  df-cnv 4611  df-dm 4613  df-rn 4614  df-iota 5152  df-fv 5195

This theorem is referenced by: (None)