Theorem elfvex 5660

Description: If a function value is inhabited, the function value is a set. (Contributed by Jim Kingdon, 30-Jan-2026.)

Assertion

Ref	Expression
elfvex	|- ( A e. ( F ` B ) -> ( F ` B ) e. _V )

Proof of Theorem elfvex

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fv 5325	. 2 |- ( F ` B ) = ( iota w B F w )
2		eliotaeu 5306	. . . 4 |- ( A e. ( iota w B F w ) -> E! w B F w )
3	2, 1	eleq2s 2324	. . 3 |- ( A e. ( F ` B ) -> E! w B F w )
4		euiotaex 5294	. . 3 |- ( E! w B F w -> ( iota w B F w ) e. _V )
5	3, 4	syl 14	. 2 |- ( A e. ( F ` B ) -> ( iota w B F w ) e. _V )
6	1, 5	eqeltrid 2316	1 |- ( A e. ( F ` B ) -> ( F ` B ) e. _V )

Syntax hints:   -> wi 4   E!weu 2077   e. wcel 2200   _Vcvv 2799   class class class wbr 4082   iotacio 5275   ` cfv 5317

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-sn 3672  df-pr 3673  df-uni 3888  df-iota 5277  df-fv 5325

This theorem is referenced by:  fvmbr  5661  wlkvtxiedgg  16042