Theorem ovmpt4g 5893

Description: Value of a function given by the maps-to notation. (This is the operation analog of fvmpt2 5504.) (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 1-Sep-2015.)

Hypothesis

Ref	Expression
ovmpt4g.3	|- F = ( x e. A , y e. B |-> C )

Assertion

Ref	Expression
ovmpt4g	|- ( ( x e. A /\ y e. B /\ C e. V ) -> ( x F y ) = C )

Distinct variable group:   x, y

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

Proof of Theorem ovmpt4g

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elisset 2700	. . 3 |- ( C e. V -> E. z z = C )
2		moeq 2859	. . . . . . 7 |- E* z z = C
3	2	a1i 9	. . . . . 6 |- ( ( x e. A /\ y e. B ) -> E* z z = C )
4		ovmpt4g.3	. . . . . . 7 |- F = ( x e. A , y e. B |-> C )
5		df-mpo 5779	. . . . . . 7 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
6	4, 5	eqtri 2160	. . . . . 6 |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
7	3, 6	ovidi 5889	. . . . 5 |- ( ( x e. A /\ y e. B ) -> ( z = C -> ( x F y ) = z ) )
8		eqeq2 2149	. . . . 5 |- ( z = C -> ( ( x F y ) = z <-> ( x F y ) = C ) )
9	7, 8	mpbidi 150	. . . 4 |- ( ( x e. A /\ y e. B ) -> ( z = C -> ( x F y ) = C ) )
10	9	exlimdv 1791	. . 3 |- ( ( x e. A /\ y e. B ) -> ( E. z z = C -> ( x F y ) = C ) )
11	1, 10	syl5 32	. 2 |- ( ( x e. A /\ y e. B ) -> ( C e. V -> ( x F y ) = C ) )
12	11	3impia 1178	1 |- ( ( x e. A /\ y e. B /\ C e. V ) -> ( x F y ) = C )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   E.wex 1468   e. wcel 1480   E*wmo 2000  (class class class)co 5774   {coprab 5775   e. cmpo 5776

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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779

This theorem is referenced by:  ovmpos  5894  ov2gf  5895  ovmpodxf  5896  ovmpodf  5902  ofmres  6034  fnmpoovd  6112  mapxpen  6742  cnmpt21  12460  cnmpt2t  12462  cnmptcom  12467