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Theorem ovi3 6216
Description: The value of an operation class abstraction. Special case. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
ovi3.1 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> S e. ( H X. H ) )
ovi3.2 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> R = S )
ovi3.3 |- F = { <. <. x , y >. , z >. | ( ( x e. ( H X. H ) /\ y e. ( H X. H ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) ) }
Assertion
Ref Expression
ovi3 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> ( <. A , B >. F <. C , D >. ) = S )
Distinct variable groups:   u, f, v, w, x, y, z, A   B, f, u, v, w, x, y, z   x, R, y, z   C, f, u, v, w, y, z   D, f, u, v, w, y, z   f, H, u, v, w, x, y, z   S, f, u, v, w, z
Allowed substitution hints:   C( x)   D( x)   R( w, v, u, f)   S( x, y)   F( x, y, z, w, v, u, f)
Proof of Theorem ovi3
StepHypRef Expression
1 ovi3.1 . . . 4 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> S e. ( H X. H ) )
2 elex 2833 . . . 4 |- ( S e. ( H X. H ) -> S e. _V )
31, 2syl 14 . . 3 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> S e. _V )
4 isset 2828 . . 3 |- ( S e. _V <-> E. z z = S )
53, 4sylib 122 . 2 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> E. z z = S )
6 nfv 1581 . . 3 |- F/ z ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) )
7 nfcv 2392 . . . . 5 |- F/_ z <. A , B >.
8 ovi3.3 . . . . . 6 |- F = { <. <. x , y >. , z >. | ( ( x e. ( H X. H ) /\ y e. ( H X. H ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) ) }
9 nfoprab3 6129 . . . . . 6 |- F/_ z { <. <. x , y >. , z >. | ( ( x e. ( H X. H ) /\ y e. ( H X. H ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) ) }
108, 9nfcxfr 2389 . . . . 5 |- F/_ z F
11 nfcv 2392 . . . . 5 |- F/_ z <. C , D >.
127, 10, 11nfov 6105 . . . 4 |- F/_ z ( <. A , B >. F <. C , D >. )
1312nfeq1 2402 . . 3 |- F/ z ( <. A , B >. F <. C , D >. ) = S
14 ovi3.2 . . . . . . 7 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> R = S )
1514eqeq2d 2250 . . . . . 6 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> ( z = R <-> z = S ) )
1615copsex4g 4382 . . . . 5 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> ( E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R ) <-> z = S ) )
17 opelxpi 4801 . . . . . 6 |- ( ( A e. H /\ B e. H ) -> <. A , B >. e. ( H X. H ) )
18 opelxpi 4801 . . . . . 6 |- ( ( C e. H /\ D e. H ) -> <. C , D >. e. ( H X. H ) )
19 nfcv 2392 . . . . . . 7 |- F/_ x <. A , B >.
20 nfcv 2392 . . . . . . 7 |- F/_ y <. A , B >.
21 nfcv 2392 . . . . . . 7 |- F/_ y <. C , D >.
22 nfv 1581 . . . . . . . 8 |- F/ x E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R )
23 nfoprab1 6127 . . . . . . . . . . 11 |- F/_ x { <. <. x , y >. , z >. | ( ( x e. ( H X. H ) /\ y e. ( H X. H ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) ) }
248, 23nfcxfr 2389 . . . . . . . . . 10 |- F/_ x F
25 nfcv 2392 . . . . . . . . . 10 |- F/_ x y
2619, 24, 25nfov 6105 . . . . . . . . 9 |- F/_ x ( <. A , B >. F y )
2726nfeq1 2402 . . . . . . . 8 |- F/ x ( <. A , B >. F y ) = z
2822, 27nfim 1625 . . . . . . 7 |- F/ x ( E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) -> ( <. A , B >. F y ) = z )
29 nfv 1581 . . . . . . . 8 |- F/ y E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R )
30 nfoprab2 6128 . . . . . . . . . . 11 |- F/_ y { <. <. x , y >. , z >. | ( ( x e. ( H X. H ) /\ y e. ( H X. H ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) ) }
318, 30nfcxfr 2389 . . . . . . . . . 10 |- F/_ y F
3220, 31, 21nfov 6105 . . . . . . . . 9 |- F/_ y ( <. A , B >. F <. C , D >. )
3332nfeq1 2402 . . . . . . . 8 |- F/ y ( <. A , B >. F <. C , D >. ) = z
3429, 33nfim 1625 . . . . . . 7 |- F/ y ( E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R ) -> ( <. A , B >. F <. C , D >. ) = z )
35 eqeq1 2245 . . . . . . . . . . 11 |- ( x = <. A , B >. -> ( x = <. w , v >. <-> <. A , B >. = <. w , v >. ) )
3635anbi1d 469 . . . . . . . . . 10 |- ( x = <. A , B >. -> ( ( x = <. w , v >. /\ y = <. u , f >. ) <-> ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) ) )
3736anbi1d 469 . . . . . . . . 9 |- ( x = <. A , B >. -> ( ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) ) )
38374exbidv 1923 . . . . . . . 8 |- ( x = <. A , B >. -> ( E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) ) )
39 oveq1 6082 . . . . . . . . 9 |- ( x = <. A , B >. -> ( x F y ) = ( <. A , B >. F y ) )
4039eqeq1d 2247 . . . . . . . 8 |- ( x = <. A , B >. -> ( ( x F y ) = z <-> ( <. A , B >. F y ) = z ) )
4138, 40imbi12d 234 . . . . . . 7 |- ( x = <. A , B >. -> ( ( E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) -> ( x F y ) = z ) <-> ( E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) -> ( <. A , B >. F y ) = z ) ) )
42 eqeq1 2245 . . . . . . . . . . 11 |- ( y = <. C , D >. -> ( y = <. u , f >. <-> <. C , D >. = <. u , f >. ) )
4342anbi2d 468 . . . . . . . . . 10 |- ( y = <. C , D >. -> ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) <-> ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) ) )
4443anbi1d 469 . . . . . . . . 9 |- ( y = <. C , D >. -> ( ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R ) ) )
45444exbidv 1923 . . . . . . . 8 |- ( y = <. C , D >. -> ( E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R ) ) )
46 oveq2 6083 . . . . . . . . 9 |- ( y = <. C , D >. -> ( <. A , B >. F y ) = ( <. A , B >. F <. C , D >. ) )
4746eqeq1d 2247 . . . . . . . 8 |- ( y = <. C , D >. -> ( ( <. A , B >. F y ) = z <-> ( <. A , B >. F <. C , D >. ) = z ) )
4845, 47imbi12d 234 . . . . . . 7 |- ( y = <. C , D >. -> ( ( E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) -> ( <. A , B >. F y ) = z ) <-> ( E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R ) -> ( <. A , B >. F <. C , D >. ) = z ) ) )
49 moeq 3001 . . . . . . . . . . . 12 |- E* z z = R
5049mosubop 4836 . . . . . . . . . . 11 |- E* z E. u E. f ( y = <. u , f >. /\ z = R )
5150mosubop 4836 . . . . . . . . . 10 |- E* z E. w E. v ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = R ) )
52 anass 405 . . . . . . . . . . . . . 14 |- ( ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> ( x = <. w , v >. /\ ( y = <. u , f >. /\ z = R ) ) )
53522exbii 1659 . . . . . . . . . . . . 13 |- ( E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> E. u E. f ( x = <. w , v >. /\ ( y = <. u , f >. /\ z = R ) ) )
54 19.42vv 1967 . . . . . . . . . . . . 13 |- ( E. u E. f ( x = <. w , v >. /\ ( y = <. u , f >. /\ z = R ) ) <-> ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = R ) ) )
5553, 54bitri 184 . . . . . . . . . . . 12 |- ( E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = R ) ) )
56552exbii 1659 . . . . . . . . . . 11 |- ( E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> E. w E. v ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = R ) ) )
5756mobii 2123 . . . . . . . . . 10 |- ( E* z E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> E* z E. w E. v ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = R ) ) )
5851, 57mpbir 146 . . . . . . . . 9 |- E* z E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R )
5958a1i 9 . . . . . . . 8 |- ( ( x e. ( H X. H ) /\ y e. ( H X. H ) ) -> E* z E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) )
6059, 8ovidi 6197 . . . . . . 7 |- ( ( x e. ( H X. H ) /\ y e. ( H X. H ) ) -> ( E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) -> ( x F y ) = z ) )
6119, 20, 21, 28, 34, 41, 48, 60vtocl2gaf 2890 . . . . . 6 |- ( ( <. A , B >. e. ( H X. H ) /\ <. C , D >. e. ( H X. H ) ) -> ( E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R ) -> ( <. A , B >. F <. C , D >. ) = z ) )
6217, 18, 61syl2an 289 . . . . 5 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> ( E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R ) -> ( <. A , B >. F <. C , D >. ) = z ) )
6316, 62sylbird 170 . . . 4 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> ( z = S -> ( <. A , B >. F <. C , D >. ) = z ) )
64 eqeq2 2248 . . . 4 |- ( z = S -> ( ( <. A , B >. F <. C , D >. ) = z <-> ( <. A , B >. F <. C , D >. ) = S ) )
6563, 64mpbidi 151 . . 3 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> ( z = S -> ( <. A , B >. F <. C , D >. ) = S ) )
666, 13, 65exlimd 1650 . 2 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> ( E. z z = S -> ( <. A , B >. F <. C , D >. ) = S ) )
675, 66mpd 13 1 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> ( <. A , B >. F <. C , D >. ) = S )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   E*wmo 2087   e. wcel 2209   _Vcvv 2821   <.cop 3708   X. cxp 4767  (class class class)co 6075   {coprab 6076
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-setind 4679
This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-oprab 6079
This theorem is referenced by:  oviec  6905  addcnsr  8191  mulcnsr  8192
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