Theorem fvmbr 5710

Description: If a function value is inhabited, the argument is related to the function value. (Contributed by Jim Kingdon, 31-Jan-2026.)

Assertion

Ref	Expression
fvmbr	|- ( A e. ( F ` X ) -> X F ( F ` X ) )

Proof of Theorem fvmbr

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fv 5365	. . 3 |- ( F ` X ) = ( iota w X F w )
2	1	eqcomi 2238	. 2 |- ( iota w X F w ) = ( F ` X )
3		elfvfvex 5709	. . 3 |- ( A e. ( F ` X ) -> ( F ` X ) e. _V )
4		eliotaeu 5346	. . . 4 |- ( A e. ( iota w X F w ) -> E! w X F w )
5	4, 1	eleq2s 2329	. . 3 |- ( A e. ( F ` X ) -> E! w X F w )
6		breq2 4118	. . . 4 |- ( w = ( F ` X ) -> ( X F w <-> X F ( F ` X ) ) )
7	6	iota2 5347	. . 3 |- ( ( ( F ` X ) e. _V /\ E! w X F w ) -> ( X F ( F ` X ) <-> ( iota w X F w ) = ( F ` X ) ) )
8	3, 5, 7	syl2anc 411	. 2 |- ( A e. ( F ` X ) -> ( X F ( F ` X ) <-> ( iota w X F w ) = ( F ` X ) ) )
9	2, 8	mpbiri 168	1 |- ( A e. ( F ` X ) -> X F ( F ` X ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   E!weu 2082   e. wcel 2205   _Vcvv 2815   class class class wbr 4114   iotacio 5315   ` cfv 5357

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365

This theorem is referenced by:  wlkvtxiedg  16466  wlkvtxiedgg  16467