Theorem mpofun 6105

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 2249	. . . . . 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 1496	. . . 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 2236	. . . . . 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 2139	. . . 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 6103	. 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 6005	. . . 4 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
11	9, 10	eqtri 2250	. . 3 |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
12	11	funeqi 5338	. 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 1393   = wceq 1395   E*wmo 2078   e. wcel 2200   Fun wfun 5311   {coprab 6001   e. cmpo 6002

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-rex 2514  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-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-fun 5319  df-oprab 6004  df-mpo 6005

This theorem is referenced by:  elmpocl  6199  ofexg  6221  mpoexxg  6354  mpoexw  6357  mpoxopn0yelv  6383