Theorem funfveu 5574

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 2259	. . . . 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 4037	. . . . 5 |- ( x = A -> ( x F y <-> A F y ) )
4	3	eubidv 2053	. . . 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 5287	. . . . 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 2585	. . 3 |- ( ( Fun F /\ x e. dom F ) -> E! y x F y )
9	5, 8	vtoclg 2824	. 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 1364   E!weu 2045   e. wcel 2167   A.wral 2475   class class class wbr 4034   dom cdm 4664   Rel wrel 4669   Fun wfun 5253

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-id 4329  df-cnv 4672  df-co 4673  df-dm 4674  df-fun 5261

This theorem is referenced by:  funfvex  5578