Theorem funfveu 5612

Description: A function has one value given an argument in its domain. (Contributed by Jim Kingdon, 29-Dec-2018.)

Assertion

Ref	Expression
funfveu	|- ( ( Fun F /\ A e. dom F ) -> E! y A F y )

Distinct variable groups:   y, A   y, F

Proof of Theorem funfveu

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2270	. . . . 5 |- ( x = A -> ( x e. dom F <-> A e. dom F ) )
2	1	anbi2d 464	. . . 4 |- ( x = A -> ( ( Fun F /\ x e. dom F ) <-> ( Fun F /\ A e. dom F ) ) )
3		breq1 4062	. . . . 5 |- ( x = A -> ( x F y <-> A F y ) )
4	3	eubidv 2063	. . . 4 |- ( x = A -> ( E! y x F y <-> E! y A F y ) )
5	2, 4	imbi12d 234	. . 3 |- ( x = A -> ( ( ( Fun F /\ x e. dom F ) -> E! y x F y ) <-> ( ( Fun F /\ A e. dom F ) -> E! y A F y ) ) )
6		dffun8 5318	. . . . 5 |- ( Fun F <-> ( Rel F /\ A. x e. dom F E! y x F y ) )
7	6	simprbi 275	. . . 4 |- ( Fun F -> A. x e. dom F E! y x F y )
8	7	r19.21bi 2596	. . 3 |- ( ( Fun F /\ x e. dom F ) -> E! y x F y )
9	5, 8	vtoclg 2838	. 2 |- ( A e. dom F -> ( ( Fun F /\ A e. dom F ) -> E! y A F y ) )
10	9	anabsi7 581	1 |- ( ( Fun F /\ A e. dom F ) -> E! y A F y )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   E!weu 2055   e. wcel 2178   A.wral 2486   class class class wbr 4059   dom cdm 4693   Rel wrel 4698   Fun wfun 5284

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-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-id 4358  df-cnv 4701  df-co 4702  df-dm 4703  df-fun 5292

This theorem is referenced by:  funfvex  5616