Theorem elfvm 5609

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 5260	. . . 4 |- ( A e. ( iota x B F x ) -> E! x B F x )
2		df-fv 5279	. . . 4 |- ( F ` B ) = ( iota x B F x )
3	1, 2	eleq2s 2300	. . 3 |- ( A e. ( F ` B ) -> E! x B F x )
4		euex 2084	. . 3 |- ( E! x B F x -> E. x B F x )
5		brm 4094	. . . 4 |- ( B F x -> E. k k e. F )
6	5	exlimiv 1621	. . 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 2266	. . 3 |- ( k = j -> ( k e. F <-> j e. F ) )
9	8	cbvexv 1942	. 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 1515   E!weu 2054   e. wcel 2176   class class class wbr 4044   iotacio 5230   ` cfv 5271

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-sn 3639  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279

This theorem is referenced by:  basm  12893