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Theorem ovi3 6055
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 2771 . . . 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 2766 . . 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 1539 . . 3 |- F/ z ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) )
7 nfcv 2336 . . . . 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 5969 . . . . . 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 2333 . . . . 5 |- F/_ z F
11 nfcv 2336 . . . . 5 |- F/_ z <. C , D >.
127, 10, 11nfov 5948 . . . 4 |- F/_ z ( <. A , B >. F <. C , D >. )
1312nfeq1 2346 . . 3 |- F/ z ( <. A , B >. F <. C , D >. ) = S
14 ovi3.2 . . . . . . 7 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> R = S )
1514eqeq2d 2205 . . . . . 6 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> ( z = R <-> z = S ) )
1615copsex4g 4276 . . . . 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 4691 . . . . . 6 |- ( ( A e. H /\ B e. H ) -> <. A , B >. e. ( H X. H ) )
18 opelxpi 4691 . . . . . 6 |- ( ( C e. H /\ D e. H ) -> <. C , D >. e. ( H X. H ) )
19 nfcv 2336 . . . . . . 7 |- F/_ x <. A , B >.
20 nfcv 2336 . . . . . . 7 |- F/_ y <. A , B >.
21 nfcv 2336 . . . . . . 7 |- F/_ y <. C , D >.
22 nfv 1539 . . . . . . . 8 |- F/ x E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R )
23 nfoprab1 5967 . . . . . . . . . . 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 2333 . . . . . . . . . 10 |- F/_ x F
25 nfcv 2336 . . . . . . . . . 10 |- F/_ x y
2619, 24, 25nfov 5948 . . . . . . . . 9 |- F/_ x ( <. A , B >. F y )
2726nfeq1 2346 . . . . . . . 8 |- F/ x ( <. A , B >. F y ) = z
2822, 27nfim 1583 . . . . . . 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 1539 . . . . . . . 8 |- F/ y E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R )
30 nfoprab2 5968 . . . . . . . . . . 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 2333 . . . . . . . . . 10 |- F/_ y F
3220, 31, 21nfov 5948 . . . . . . . . 9 |- F/_ y ( <. A , B >. F <. C , D >. )
3332nfeq1 2346 . . . . . . . 8 |- F/ y ( <. A , B >. F <. C , D >. ) = z
3429, 33nfim 1583 . . . . . . 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 2200 . . . . . . . . . . 11 |- ( x = <. A , B >. -> ( x = <. w , v >. <-> <. A , B >. = <. w , v >. ) )
3635anbi1d 465 . . . . . . . . . 10 |- ( x = <. A , B >. -> ( ( x = <. w , v >. /\ y = <. u , f >. ) <-> ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) ) )
3736anbi1d 465 . . . . . . . . 9 |- ( x = <. A , B >. -> ( ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) ) )
38374exbidv 1881 . . . . . . . 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 5925 . . . . . . . . 9 |- ( x = <. A , B >. -> ( x F y ) = ( <. A , B >. F y ) )
4039eqeq1d 2202 . . . . . . . 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 2200 . . . . . . . . . . 11 |- ( y = <. C , D >. -> ( y = <. u , f >. <-> <. C , D >. = <. u , f >. ) )
4342anbi2d 464 . . . . . . . . . 10 |- ( y = <. C , D >. -> ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) <-> ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) ) )
4443anbi1d 465 . . . . . . . . 9 |- ( y = <. C , D >. -> ( ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R ) ) )
45444exbidv 1881 . . . . . . . 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 5926 . . . . . . . . 9 |- ( y = <. C , D >. -> ( <. A , B >. F y ) = ( <. A , B >. F <. C , D >. ) )
4746eqeq1d 2202 . . . . . . . 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 2935 . . . . . . . . . . . 12 |- E* z z = R
5049mosubop 4725 . . . . . . . . . . 11 |- E* z E. u E. f ( y = <. u , f >. /\ z = R )
5150mosubop 4725 . . . . . . . . . 10 |- E* z E. w E. v ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = R ) )
52 anass 401 . . . . . . . . . . . . . 14 |- ( ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> ( x = <. w , v >. /\ ( y = <. u , f >. /\ z = R ) ) )
53522exbii 1617 . . . . . . . . . . . . 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 1923 . . . . . . . . . . . . 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 1617 . . . . . . . . . . 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 2079 . . . . . . . . . 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 6037 . . . . . . 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 2827 . . . . . 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 2203 . . . 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 1608 . 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 1364   E.wex 1503   E*wmo 2043   e. wcel 2164   _Vcvv 2760   <.cop 3621   X. cxp 4657  (class class class)co 5918   {coprab 5919
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4569
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922
This theorem is referenced by:  oviec  6695  addcnsr  7894  mulcnsr  7895
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