Theorem mpofun 6037

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 2224	. . . . . 6 |- ( ( z = C /\ w = C ) -> z = w )
2	1	ad2ant2l 508	. . . . 5 |- ( ( ( ( x e. A /\ y e. B ) /\ z = C ) /\ ( ( x e. A /\ y e. B ) /\ w = C ) ) -> z = w )
3	2	gen2 1472	. . . 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 2211	. . . . . 6 |- ( z = w -> ( z = C <-> w = C ) )
5	4	anbi2d 464	. . . . 5 |- ( z = w -> ( ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( ( x e. A /\ y e. B ) /\ w = C ) ) )
6	5	mo4 2114	. . . 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 146	. . 3 |- E* z ( ( x e. A /\ y e. B ) /\ z = C )
8	7	funoprab 6035	. 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 5939	. . . 4 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
11	9, 10	eqtri 2225	. . 3 |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
12	11	funeqi 5289	. 2 |- ( Fun F <-> Fun { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) } )
13	8, 12	mpbir 146	1 |- Fun F

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1370   = wceq 1372   E*wmo 2054   e. wcel 2175   Fun wfun 5262   {coprab 5935   e. cmpo 5936

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-fun 5270  df-oprab 5938  df-mpo 5939

This theorem is referenced by:  elmpocl  6131  ofexg  6153  mpoexxg  6286  mpoexw  6289  mpoxopn0yelv  6315