Theorem mpofun 5955

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 2190	. . . . . 6 |- ( ( z = C /\ w = C ) -> z = w )
2	1	ad2ant2l 505	. . . . 5 |- ( ( ( ( x e. A /\ y e. B ) /\ z = C ) /\ ( ( x e. A /\ y e. B ) /\ w = C ) ) -> z = w )
3	2	gen2 1443	. . . 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 2177	. . . . . 6 |- ( z = w -> ( z = C <-> w = C ) )
5	4	anbi2d 461	. . . . 5 |- ( z = w -> ( ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( ( x e. A /\ y e. B ) /\ w = C ) ) )
6	5	mo4 2080	. . . 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 5953	. 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 5858	. . . 4 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
11	9, 10	eqtri 2191	. . 3 |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
12	11	funeqi 5219	. 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 1346   = wceq 1348   E*wmo 2020   e. wcel 2141   Fun wfun 5192   {coprab 5854   e. cmpo 5855

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-fun 5200  df-oprab 5857  df-mpo 5858

This theorem is referenced by:  elmpocl  6047  ofexg  6065  mpoexxg  6189  mpoexw  6192  mpoxopn0yelv  6218