Theorem relrnfvex 5713

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 5711	. 2 |- ( Rel F -> ( F ` A ) C_ U. ran F )
2		uniexg 4585	. 2 |- ( ran F e. _V -> U. ran F e. _V )
3		ssexg 4272	. 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209   _Vcvv 2821   C_ wss 3220   U.cuni 3935   ran crn 4775   Rel wrel 4779   ` cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-cnv 4782  df-dm 4784  df-rn 4785  df-iota 5337  df-fv 5385

This theorem is used by: (None)