Theorem ovmpt4g 5643

Description: Value of a function given by the "maps to" notation. (This is the operation analog of fvmpt2 5275.) (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 2613	. . 3 |- ( C e. V -> E. z z = C )
2		moeq 2767	. . . . . . 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-mpt2 5537	. . . . . . 7 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
6	4, 5	eqtri 2101	. . . . . 6 |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
7	3, 6	ovidi 5639	. . . . 5 |- ( ( x e. A /\ y e. B ) -> ( z = C -> ( x F y ) = z ) )
8		eqeq2 2090	. . . . 5 |- ( z = C -> ( ( x F y ) = z <-> ( x F y ) = C ) )
9	7, 8	mpbidi 149	. . . 4 |- ( ( x e. A /\ y e. B ) -> ( z = C -> ( x F y ) = C ) )
10	9	exlimdv 1740	. . 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 1135	1 |- ( ( x e. A /\ y e. B /\ C e. V ) -> ( x F y ) = C )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   = wceq 1284   E.wex 1421   e. wcel 1433   E*wmo 1942  (class class class)co 5532   {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-in1 576  ax-in2 577  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  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  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-ne 2246  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537

This theorem is referenced by:  ovmpt2s  5644  ov2gf  5645  ovmpt2dxf  5646  ovmpt2df  5652  ofmres  5783