Theorem mpt2fun 5623

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

Hypothesis

Ref	Expression
mpt2fun.1	|- F = ( x e. A , y e. B |-> C )

Assertion

Ref	Expression
mpt2fun	|- 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 mpt2fun

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

Step	Hyp	Ref	Expression
1		eqtr3 2100	. . . . . 6 |- ( ( z = C /\ w = C ) -> z = w )
2	1	ad2ant2l 491	. . . . 5 |- ( ( ( ( x e. A /\ y e. B ) /\ z = C ) /\ ( ( x e. A /\ y e. B ) /\ w = C ) ) -> z = w )
3	2	gen2 1379	. . . 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 2087	. . . . . 6 |- ( z = w -> ( z = C <-> w = C ) )
5	4	anbi2d 451	. . . . 5 |- ( z = w -> ( ( ( x e. A /\ y e. B ) /\ z = C ) <-> ( ( x e. A /\ y e. B ) /\ w = C ) ) )
6	5	mo4 2002	. . . 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 144	. . 3 |- E* z ( ( x e. A /\ y e. B ) /\ z = C )
8	7	funoprab 5621	. 2 |- Fun { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
9		mpt2fun.1	. . . 4 |- F = ( x e. A , y e. B |-> C )
10		df-mpt2 5537	. . . 4 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
11	9, 10	eqtri 2101	. . 3 |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
12	11	funeqi 4942	. 2 |- ( Fun F <-> Fun { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) } )
13	8, 12	mpbir 144	1 |- Fun F

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   = wceq 1284   e. wcel 1433   E*wmo 1942   Fun wfun 4916   {coprab 5533   |-> cmpt2 5534

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-fun 4924  df-oprab 5536  df-mpt2 5537

This theorem is referenced by:  elmpt2cl  5718  ofexg  5736  mpt2exxg  5853  mpt2xopn0yelv  5877