Theorem mptmex 5939

Description: If a function given by maps-to notation is inhabited, then the class it is defined on is inhabited. (Contributed by Jim Kingdon, 22-Jul-2026.)

Assertion

Ref	Expression
mptmex	|- ( C e. ( x e. A |-> B ) -> E. y y e. A )

Distinct variable group:   x, A, y

Allowed substitution hints:   B( x, y)   C( x, y)

Proof of Theorem mptmex

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex2 2838	. . . . 5 |- ( C e. { <. x , z >. | ( x e. A /\ z = B ) } -> E. w w e. { <. x , z >. | ( x e. A /\ z = B ) } )
2		opabm 4421	. . . . 5 |- ( E. w w e. { <. x , z >. | ( x e. A /\ z = B ) } <-> E. x E. z ( x e. A /\ z = B ) )
3	1, 2	sylib 122	. . . 4 |- ( C e. { <. x , z >. | ( x e. A /\ z = B ) } -> E. x E. z ( x e. A /\ z = B ) )
4		df-mpt 4192	. . . 4 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
5	3, 4	eleq2s 2333	. . 3 |- ( C e. ( x e. A |-> B ) -> E. x E. z ( x e. A /\ z = B ) )
6		simpl 109	. . . . 5 |- ( ( x e. A /\ z = B ) -> x e. A )
7	6	exlimiv 1651	. . . 4 |- ( E. z ( x e. A /\ z = B ) -> x e. A )
8	7	eximi 1653	. . 3 |- ( E. x E. z ( x e. A /\ z = B ) -> E. x x e. A )
9	5, 8	syl 14	. 2 |- ( C e. ( x e. A |-> B ) -> E. x x e. A )
10		eleq1w 2299	. . 3 |- ( x = y -> ( x e. A <-> y e. A ) )
11	10	cbvexv 1974	. 2 |- ( E. x x e. A <-> E. y y e. A )
12	9, 11	sylib 122	1 |- ( C e. ( x e. A |-> B ) -> E. y y e. A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   {copab 4189   |-> cmpt 4190

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-opab 4191  df-mpt 4192

This theorem is referenced by:  asclfval  15004