Theorem funfveu 5652

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 2294	. . . . 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 4091	. . . . 5 |- ( x = A -> ( x F y <-> A F y ) )
4	3	eubidv 2087	. . . 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 5354	. . . . 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 2620	. . 3 |- ( ( Fun F /\ x e. dom F ) -> E! y x F y )
9	5, 8	vtoclg 2864	. 2 |- ( A e. dom F -> ( ( Fun F /\ A e. dom F ) -> E! y A F y ) )
10	9	anabsi7 583	1 |- ( ( Fun F /\ A e. dom F ) -> E! y A F y )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   E!weu 2079   e. wcel 2202   A.wral 2510   class class class wbr 4088   dom cdm 4725   Rel wrel 4730   Fun wfun 5320

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-cnv 4733  df-co 4734  df-dm 4735  df-fun 5328

This theorem is referenced by:  funfvex  5656