Theorem funfveu 5499

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 2229	. . . . 5 |- ( x = A -> ( x e. dom F <-> A e. dom F ) )
2	1	anbi2d 460	. . . 4 |- ( x = A -> ( ( Fun F /\ x e. dom F ) <-> ( Fun F /\ A e. dom F ) ) )
3		breq1 3985	. . . . 5 |- ( x = A -> ( x F y <-> A F y ) )
4	3	eubidv 2022	. . . 4 |- ( x = A -> ( E! y x F y <-> E! y A F y ) )
5	2, 4	imbi12d 233	. . 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 5216	. . . . 5 |- ( Fun F <-> ( Rel F /\ A. x e. dom F E! y x F y ) )
7	6	simprbi 273	. . . 4 |- ( Fun F -> A. x e. dom F E! y x F y )
8	7	r19.21bi 2554	. . 3 |- ( ( Fun F /\ x e. dom F ) -> E! y x F y )
9	5, 8	vtoclg 2786	. 2 |- ( A e. dom F -> ( ( Fun F /\ A e. dom F ) -> E! y A F y ) )
10	9	anabsi7 571	1 |- ( ( Fun F /\ A e. dom F ) -> E! y A F y )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   E!weu 2014   e. wcel 2136   A.wral 2444   class class class wbr 3982   dom cdm 4604   Rel wrel 4609   Fun wfun 5182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-cnv 4612  df-co 4613  df-dm 4614  df-fun 5190

This theorem is referenced by:  funfvex  5503