Theorem elfvm 5632

Description: If a function value has a member, the function is inhabited. (Contributed by Jim Kingdon, 14-Jun-2025.)

Assertion

Ref	Expression
elfvm	|- ( A e. ( F ` B ) -> E. j j e. F )

Distinct variable group:   j, F

Allowed substitution hints:   A( j)   B( j)

Proof of Theorem elfvm

Dummy variables k x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliotaeu 5279	. . . 4 |- ( A e. ( iota x B F x ) -> E! x B F x )
2		df-fv 5298	. . . 4 |- ( F ` B ) = ( iota x B F x )
3	1, 2	eleq2s 2302	. . 3 |- ( A e. ( F ` B ) -> E! x B F x )
4		euex 2085	. . 3 |- ( E! x B F x -> E. x B F x )
5		brm 4110	. . . 4 |- ( B F x -> E. k k e. F )
6	5	exlimiv 1622	. . 3 |- ( E. x B F x -> E. k k e. F )
7	3, 4, 6	3syl 17	. 2 |- ( A e. ( F ` B ) -> E. k k e. F )
8		eleq1w 2268	. . 3 |- ( k = j -> ( k e. F <-> j e. F ) )
9	8	cbvexv 1943	. 2 |- ( E. k k e. F <-> E. j j e. F )
10	7, 9	sylib 122	1 |- ( A e. ( F ` B ) -> E. j j e. F )

Syntax hints:   -> wi 4   E.wex 1516   E!weu 2055   e. wcel 2178   class class class wbr 4059   iotacio 5249   ` cfv 5290

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-sn 3649  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298

This theorem is referenced by:  basm  13008