Theorem ovmpt4g 5964

Description: Value of a function given by the maps-to notation. (This is the operation analog of fvmpt2 5569.) (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 2740	. . 3 |- ( C e. V -> E. z z = C )
2		moeq 2901	. . . . . . 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 5847	. . . . . . 7 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
6	4, 5	eqtri 2186	. . . . . 6 |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
7	3, 6	ovidi 5960	. . . . 5 |- ( ( x e. A /\ y e. B ) -> ( z = C -> ( x F y ) = z ) )
8		eqeq2 2175	. . . . 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 1807	. . 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 1190	1 |- ( ( x e. A /\ y e. B /\ C e. V ) -> ( x F y ) = C )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   E.wex 1480   E*wmo 2015   e. wcel 2136  (class class class)co 5842   {coprab 5843   e. cmpo 5844

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847

This theorem is referenced by:  ovmpos  5965  ov2gf  5966  ovmpodxf  5967  ovmpodf  5973  ofmres  6104  fnmpoovd  6183  mapxpen  6814  cnmpt21  12931  cnmpt2t  12933  cnmptcom  12938