Theorem mpofun 5881

Description: The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.)

Hypothesis

Ref	Expression
mpofun.1	|- F = ( x e. A , y e. B |-> C )

Assertion

Ref	Expression
mpofun	|- Fun F

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)   B( x, y)   C( x, y)   F( x, y)

Proof of Theorem mpofun

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqtr3 2160	. . . . . 6 |- ( ( z = C /\ w = C ) -> z = w )
2	1	ad2ant2l 500	. . . . 5 |- ( ( ( ( x e. A /\ y e. B ) /\ z = C ) /\ ( ( x e. A /\ y e. B ) /\ w = C ) ) -> z = w )
3	2	gen2 1427	. . . 4 |- A. z A. w ( ( ( ( x e. A /\ y e. B ) /\ z = C ) /\ ( ( x e. A /\ y e. B ) /\ w = C ) ) -> z = w )
4		eqeq1 2147	. . . . . 6 |- ( z = w -> ( z = C <-> w = C ) )
5	4	anbi2d 460	. . . . 5 |- ( z = w -> ( ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( ( x e. A /\ y e. B ) /\ w = C ) ) )
6	5	mo4 2061	. . . 4 |- ( E* z ( ( x e. A /\ y e. B ) /\ z = C ) <-> A. z A. w ( ( ( ( x e. A /\ y e. B ) /\ z = C ) /\ ( ( x e. A /\ y e. B ) /\ w = C ) ) -> z = w ) )
7	3, 6	mpbir 145	. . 3 |- E* z ( ( x e. A /\ y e. B ) /\ z = C )
8	7	funoprab 5879	. 2 |- Fun { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
9		mpofun.1	. . . 4 |- F = ( x e. A , y e. B |-> C )
10		df-mpo 5787	. . . 4 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
11	9, 10	eqtri 2161	. . 3 |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
12	11	funeqi 5152	. 2 |- ( Fun F <-> Fun { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) } )
13	8, 12	mpbir 145	1 |- Fun F

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1330   = wceq 1332   e. wcel 1481   E*wmo 2001   Fun wfun 5125   {coprab 5783   e. cmpo 5784

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-fun 5133  df-oprab 5786  df-mpo 5787

This theorem is referenced by:  elmpocl  5976  ofexg  5994  mpoexxg  6116  mpoxopn0yelv  6144