Theorem funfveu 5639

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 2292	. . . . 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 4085	. . . . 5 |- ( x = A -> ( x F y <-> A F y ) )
4	3	eubidv 2085	. . . 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 5345	. . . . 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 2618	. . 3 |- ( ( Fun F /\ x e. dom F ) -> E! y x F y )
9	5, 8	vtoclg 2861	. 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 1395   E!weu 2077   e. wcel 2200   A.wral 2508   class class class wbr 4082   dom cdm 4718   Rel wrel 4723   Fun wfun 5311

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-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-id 4383  df-cnv 4726  df-co 4727  df-dm 4728  df-fun 5319

This theorem is referenced by:  funfvex  5643