Theorem ovmpt4g 6045

Description: Value of a function given by the maps-to notation. (This is the operation analog of fvmpt2 5645.) (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 2777	. . 3 |- ( C e. V -> E. z z = C )
2		moeq 2939	. . . . . . 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 5927	. . . . . . 7 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
6	4, 5	eqtri 2217	. . . . . 6 |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
7	3, 6	ovidi 6041	. . . . 5 |- ( ( x e. A /\ y e. B ) -> ( z = C -> ( x F y ) = z ) )
8		eqeq2 2206	. . . . 5 |- ( z = C -> ( ( x F y ) = z <-> ( x F y ) = C ) )
9	7, 8	mpbidi 151	. . . 4 |- ( ( x e. A /\ y e. B ) -> ( z = C -> ( x F y ) = C ) )
10	9	exlimdv 1833	. . 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 1202	1 |- ( ( x e. A /\ y e. B /\ C e. V ) -> ( x F y ) = C )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   E.wex 1506   E*wmo 2046   e. wcel 2167  (class class class)co 5922   {coprab 5923   e. cmpo 5924

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-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927

This theorem is referenced by:  ovmpos  6046  ov2gf  6047  ovmpodxf  6048  ovmpodf  6054  ofmres  6193  fnmpoovd  6273  mapxpen  6909  cnmpt21  14527  cnmpt2t  14529  cnmptcom  14534