Theorem ovmpt4g 6783

Description: Value of a function given by the "maps to" notation. (This is the operation analogue of fvmpt2 6291.) (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 3215	. . 3 |- ( C e. V -> E. z z = C )
2		moeq 3382	. . . . . . 7 |- E* z z = C
3	2	a1i 11	. . . . . 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 6655	. . . . . . 7 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
6	4, 5	eqtri 2644	. . . . . 6 |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
7	3, 6	ovidi 6779	. . . . 5 |- ( ( x e. A /\ y e. B ) -> ( z = C -> ( x F y ) = z ) )
8		eqeq2 2633	. . . . 5 |- ( z = C -> ( ( x F y ) = z <-> ( x F y ) = C ) )
9	7, 8	mpbidi 231	. . . 4 |- ( ( x e. A /\ y e. B ) -> ( z = C -> ( x F y ) = C ) )
10	9	exlimdv 1861	. . 3 |- ( ( x e. A /\ y e. B ) -> ( E. z z = C -> ( x F y ) = C ) )
11	1, 10	syl5 34	. 2 |- ( ( x e. A /\ y e. B ) -> ( C e. V -> ( x F y ) = C ) )
12	11	3impia 1261	1 |- ( ( x e. A /\ y e. B /\ C e. V ) -> ( x F y ) = C )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E*wmo 2471  (class class class)co 6650   {coprab 6651   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  ovmpt2s  6784  ov2gf  6785  ovmpt2dxf  6786  ovmpt2df  6792  ofmres  7164  fnmpt2ovd  7252  mapxpen  8126  pwfseqlem2  9481  pwfseqlem3  9482  fullfunc  16566  fthfunc  16567  prfcl  16843  curf1cl  16868  curfcl  16872  hofcl  16899  gsum2d2lem  18372  gsum2d2  18373  gsumcom2  18374  dprdval  18402  dprd2d2  18443  cnmpt21  21474  cnmpt2t  21476  cnmptcom  21481  cnmpt2k  21491  xkocnv  21617  madjusmdetlem1  29893  madjusmdetlem3  29895  finxpreclem5  33232  sdclem2  33538  smflimlem1  40979  smflimlem2  40980  aovmpt4g  41281  ovmpt2rdxf  42117