Theorem elfvm 5660

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 5307	. . . 4 |- ( A e. ( iota x B F x ) -> E! x B F x )
2		df-fv 5326	. . . 4 |- ( F ` B ) = ( iota x B F x )
3	1, 2	eleq2s 2324	. . 3 |- ( A e. ( F ` B ) -> E! x B F x )
4		euex 2107	. . 3 |- ( E! x B F x -> E. x B F x )
5		brm 4134	. . . 4 |- ( B F x -> E. k k e. F )
6	5	exlimiv 1644	. . 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 2290	. . 3 |- ( k = j -> ( k e. F <-> j e. F ) )
9	8	cbvexv 1965	. 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 1538   E!weu 2077   e. wcel 2200   class class class wbr 4083   iotacio 5276   ` cfv 5318

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-sn 3672  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326

This theorem is referenced by:  basm  13094