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Theorem isalg 25824
Description: The predicate "has the structure required by  Ded and  Cat OLD." (Contributed by FL, 24-Oct-2007.)
Hypotheses
Ref Expression
isalg.1  |-  M  =  dom  D
isalg.2  |-  O  =  dom  J
Assertion
Ref Expression
isalg  |-  ( ( ( D  e.  A  /\  C  e.  B  /\  J  e.  F
)  /\  R  e.  G )  ->  ( <. <. D ,  C >. ,  <. J ,  R >. >.  e.  Alg  <->  ( ( D : M --> O  /\  C : M --> O  /\  J : O --> M )  /\  ( Fun  R  /\  dom  R  C_  ( M  X.  M )  /\  ran  R  C_  M )
) ) )

Proof of Theorem isalg
Dummy variables  c 
d  j  r  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-alg 25819 . . 3  |-  Alg  =  { x  |  E. d E. c E. j E. r ( x  = 
<. <. d ,  c
>. ,  <. j ,  r >. >.  /\  ( d : dom  d --> dom  j  /\  c : dom  d --> dom  j  /\  j : dom  j --> dom  d
)  /\  ( Fun  r  /\  dom  r  C_  ( dom  d  X.  dom  d )  /\  ran  r  C_  dom  d ) ) }
21eleq2i 2360 . 2  |-  ( <. <. D ,  C >. , 
<. J ,  R >. >.  e.  Alg  <->  <. <. D ,  C >. ,  <. J ,  R >. >.  e.  { x  |  E. d E. c E. j E. r ( x  =  <. <. d ,  c >. ,  <. j ,  r >. >.  /\  (
d : dom  d --> dom  j  /\  c : dom  d --> dom  j  /\  j : dom  j --> dom  d )  /\  ( Fun  r  /\  dom  r  C_  ( dom  d  X. 
dom  d )  /\  ran  r  C_  dom  d
) ) } )
3 3anass 938 . . . . . . 7  |-  ( ( x  =  <. <. d ,  c >. ,  <. j ,  r >. >.  /\  (
d : dom  d --> dom  j  /\  c : dom  d --> dom  j  /\  j : dom  j --> dom  d )  /\  ( Fun  r  /\  dom  r  C_  ( dom  d  X. 
dom  d )  /\  ran  r  C_  dom  d
) )  <->  ( x  =  <. <. d ,  c
>. ,  <. j ,  r >. >.  /\  ( (
d : dom  d --> dom  j  /\  c : dom  d --> dom  j  /\  j : dom  j --> dom  d )  /\  ( Fun  r  /\  dom  r  C_  ( dom  d  X. 
dom  d )  /\  ran  r  C_  dom  d
) ) ) )
43exbii 1572 . . . . . 6  |-  ( E. r ( x  = 
<. <. d ,  c
>. ,  <. j ,  r >. >.  /\  ( d : dom  d --> dom  j  /\  c : dom  d --> dom  j  /\  j : dom  j --> dom  d
)  /\  ( Fun  r  /\  dom  r  C_  ( dom  d  X.  dom  d )  /\  ran  r  C_  dom  d ) )  <->  E. r ( x  =  <. <. d ,  c
>. ,  <. j ,  r >. >.  /\  ( (
d : dom  d --> dom  j  /\  c : dom  d --> dom  j  /\  j : dom  j --> dom  d )  /\  ( Fun  r  /\  dom  r  C_  ( dom  d  X. 
dom  d )  /\  ran  r  C_  dom  d
) ) ) )
543exbii 1574 . . . . 5  |-  ( E. d E. c E. j E. r ( x  =  <. <. d ,  c >. ,  <. j ,  r >. >.  /\  (
d : dom  d --> dom  j  /\  c : dom  d --> dom  j  /\  j : dom  j --> dom  d )  /\  ( Fun  r  /\  dom  r  C_  ( dom  d  X. 
dom  d )  /\  ran  r  C_  dom  d
) )  <->  E. d E. c E. j E. r ( x  = 
<. <. d ,  c
>. ,  <. j ,  r >. >.  /\  ( (
d : dom  d --> dom  j  /\  c : dom  d --> dom  j  /\  j : dom  j --> dom  d )  /\  ( Fun  r  /\  dom  r  C_  ( dom  d  X. 
dom  d )  /\  ran  r  C_  dom  d
) ) ) )
65abbii 2408 . . . 4  |-  { x  |  E. d E. c E. j E. r ( x  =  <. <. d ,  c >. ,  <. j ,  r >. >.  /\  (
d : dom  d --> dom  j  /\  c : dom  d --> dom  j  /\  j : dom  j --> dom  d )  /\  ( Fun  r  /\  dom  r  C_  ( dom  d  X. 
dom  d )  /\  ran  r  C_  dom  d
) ) }  =  { x  |  E. d E. c E. j E. r ( x  = 
<. <. d ,  c
>. ,  <. j ,  r >. >.  /\  ( (
d : dom  d --> dom  j  /\  c : dom  d --> dom  j  /\  j : dom  j --> dom  d )  /\  ( Fun  r  /\  dom  r  C_  ( dom  d  X. 
dom  d )  /\  ran  r  C_  dom  d
) ) ) }
76eleq2i 2360 . . 3  |-  ( <. <. D ,  C >. , 
<. J ,  R >. >.  e.  { x  |  E. d E. c E. j E. r ( x  = 
<. <. d ,  c
>. ,  <. j ,  r >. >.  /\  ( d : dom  d --> dom  j  /\  c : dom  d --> dom  j  /\  j : dom  j --> dom  d
)  /\  ( Fun  r  /\  dom  r  C_  ( dom  d  X.  dom  d )  /\  ran  r  C_  dom  d ) ) }  <->  <. <. D ,  C >. ,  <. J ,  R >. >.  e.  { x  |  E. d E. c E. j E. r ( x  =  <. <. d ,  c >. ,  <. j ,  r >. >.  /\  (
( d : dom  d
--> dom  j  /\  c : dom  d --> dom  j  /\  j : dom  j --> dom  d )  /\  ( Fun  r  /\  dom  r  C_  ( dom  d  X. 
dom  d )  /\  ran  r  C_  dom  d
) ) ) } )
8 feq1 5391 . . . . . . 7  |-  ( d  =  D  ->  (
d : dom  d --> dom  j  <->  D : dom  d --> dom  j ) )
9 dmeq 4895 . . . . . . . . 9  |-  ( d  =  D  ->  dom  d  =  dom  D )
10 isalg.1 . . . . . . . . 9  |-  M  =  dom  D
119, 10syl6eqr 2346 . . . . . . . 8  |-  ( d  =  D  ->  dom  d  =  M )
1211feq2d 5396 . . . . . . 7  |-  ( d  =  D  ->  ( D : dom  d --> dom  j  <->  D : M --> dom  j
) )
138, 12bitrd 244 . . . . . 6  |-  ( d  =  D  ->  (
d : dom  d --> dom  j  <->  D : M --> dom  j
) )
1411feq2d 5396 . . . . . 6  |-  ( d  =  D  ->  (
c : dom  d --> dom  j  <->  c : M --> dom  j ) )
15 feq3 5393 . . . . . . 7  |-  ( dom  d  =  M  -> 
( j : dom  j
--> dom  d  <->  j : dom  j --> M ) )
1611, 15syl 15 . . . . . 6  |-  ( d  =  D  ->  (
j : dom  j --> dom  d  <->  j : dom  j
--> M ) )
1713, 14, 163anbi123d 1252 . . . . 5  |-  ( d  =  D  ->  (
( d : dom  d
--> dom  j  /\  c : dom  d --> dom  j  /\  j : dom  j --> dom  d )  <->  ( D : M --> dom  j  /\  c : M --> dom  j  /\  j : dom  j --> M ) ) )
1811, 11xpeq12d 4730 . . . . . . 7  |-  ( d  =  D  ->  ( dom  d  X.  dom  d
)  =  ( M  X.  M ) )
1918sseq2d 3219 . . . . . 6  |-  ( d  =  D  ->  ( dom  r  C_  ( dom  d  X.  dom  d
)  <->  dom  r  C_  ( M  X.  M ) ) )
2011sseq2d 3219 . . . . . 6  |-  ( d  =  D  ->  ( ran  r  C_  dom  d  <->  ran  r  C_  M )
)
2119, 203anbi23d 1255 . . . . 5  |-  ( d  =  D  ->  (
( Fun  r  /\  dom  r  C_  ( dom  d  X.  dom  d
)  /\  ran  r  C_  dom  d )  <->  ( Fun  r  /\  dom  r  C_  ( M  X.  M
)  /\  ran  r  C_  M ) ) )
2217, 21anbi12d 691 . . . 4  |-  ( d  =  D  ->  (
( ( d : dom  d --> dom  j  /\  c : dom  d --> dom  j  /\  j : dom  j --> dom  d
)  /\  ( Fun  r  /\  dom  r  C_  ( dom  d  X.  dom  d )  /\  ran  r  C_  dom  d ) )  <->  ( ( D : M --> dom  j  /\  c : M --> dom  j  /\  j : dom  j --> M )  /\  ( Fun  r  /\  dom  r  C_  ( M  X.  M
)  /\  ran  r  C_  M ) ) ) )
23 feq1 5391 . . . . . 6  |-  ( c  =  C  ->  (
c : M --> dom  j  <->  C : M --> dom  j
) )
24233anbi2d 1257 . . . . 5  |-  ( c  =  C  ->  (
( D : M --> dom  j  /\  c : M --> dom  j  /\  j : dom  j --> M )  <->  ( D : M
--> dom  j  /\  C : M --> dom  j  /\  j : dom  j --> M ) ) )
2524anbi1d 685 . . . 4  |-  ( c  =  C  ->  (
( ( D : M
--> dom  j  /\  c : M --> dom  j  /\  j : dom  j --> M )  /\  ( Fun  r  /\  dom  r  C_  ( M  X.  M
)  /\  ran  r  C_  M ) )  <->  ( ( D : M --> dom  j  /\  C : M --> dom  j  /\  j : dom  j --> M )  /\  ( Fun  r  /\  dom  r  C_  ( M  X.  M
)  /\  ran  r  C_  M ) ) ) )
26 dmeq 4895 . . . . . . . 8  |-  ( j  =  J  ->  dom  j  =  dom  J )
27 isalg.2 . . . . . . . 8  |-  O  =  dom  J
2826, 27syl6eqr 2346 . . . . . . 7  |-  ( j  =  J  ->  dom  j  =  O )
29 feq3 5393 . . . . . . 7  |-  ( dom  j  =  O  -> 
( D : M --> dom  j  <->  D : M --> O ) )
3028, 29syl 15 . . . . . 6  |-  ( j  =  J  ->  ( D : M --> dom  j  <->  D : M --> O ) )
31 feq3 5393 . . . . . . 7  |-  ( dom  j  =  O  -> 
( C : M --> dom  j  <->  C : M --> O ) )
3228, 31syl 15 . . . . . 6  |-  ( j  =  J  ->  ( C : M --> dom  j  <->  C : M --> O ) )
33 feq1 5391 . . . . . . 7  |-  ( j  =  J  ->  (
j : dom  j --> M 
<->  J : dom  j --> M ) )
3428feq2d 5396 . . . . . . 7  |-  ( j  =  J  ->  ( J : dom  j --> M  <-> 
J : O --> M ) )
3533, 34bitrd 244 . . . . . 6  |-  ( j  =  J  ->  (
j : dom  j --> M 
<->  J : O --> M ) )
3630, 32, 353anbi123d 1252 . . . . 5  |-  ( j  =  J  ->  (
( D : M --> dom  j  /\  C : M
--> dom  j  /\  j : dom  j --> M )  <-> 
( D : M --> O  /\  C : M --> O  /\  J : O --> M ) ) )
3736anbi1d 685 . . . 4  |-  ( j  =  J  ->  (
( ( D : M
--> dom  j  /\  C : M --> dom  j  /\  j : dom  j --> M )  /\  ( Fun  r  /\  dom  r  C_  ( M  X.  M
)  /\  ran  r  C_  M ) )  <->  ( ( D : M --> O  /\  C : M --> O  /\  J : O --> M )  /\  ( Fun  r  /\  dom  r  C_  ( M  X.  M )  /\  ran  r  C_  M ) ) ) )
38 funeq 5290 . . . . . 6  |-  ( r  =  R  ->  ( Fun  r  <->  Fun  R ) )
39 dmeq 4895 . . . . . . 7  |-  ( r  =  R  ->  dom  r  =  dom  R )
4039sseq1d 3218 . . . . . 6  |-  ( r  =  R  ->  ( dom  r  C_  ( M  X.  M )  <->  dom  R  C_  ( M  X.  M
) ) )
41 rneq 4920 . . . . . . 7  |-  ( r  =  R  ->  ran  r  =  ran  R )
4241sseq1d 3218 . . . . . 6  |-  ( r  =  R  ->  ( ran  r  C_  M  <->  ran  R  C_  M ) )
4338, 40, 423anbi123d 1252 . . . . 5  |-  ( r  =  R  ->  (
( Fun  r  /\  dom  r  C_  ( M  X.  M )  /\  ran  r  C_  M )  <-> 
( Fun  R  /\  dom  R  C_  ( M  X.  M )  /\  ran  R 
C_  M ) ) )
4443anbi2d 684 . . . 4  |-  ( r  =  R  ->  (
( ( D : M
--> O  /\  C : M
--> O  /\  J : O
--> M )  /\  ( Fun  r  /\  dom  r  C_  ( M  X.  M
)  /\  ran  r  C_  M ) )  <->  ( ( D : M --> O  /\  C : M --> O  /\  J : O --> M )  /\  ( Fun  R  /\  dom  R  C_  ( M  X.  M )  /\  ran  R  C_  M )
) ) )
4522, 25, 37, 44elo 25144 . . 3  |-  ( ( ( D  e.  A  /\  C  e.  B  /\  J  e.  F
)  /\  R  e.  G )  ->  ( <. <. D ,  C >. ,  <. J ,  R >. >.  e.  { x  |  E. d E. c E. j E. r ( x  =  <. <. d ,  c >. ,  <. j ,  r >. >.  /\  (
( d : dom  d
--> dom  j  /\  c : dom  d --> dom  j  /\  j : dom  j --> dom  d )  /\  ( Fun  r  /\  dom  r  C_  ( dom  d  X. 
dom  d )  /\  ran  r  C_  dom  d
) ) ) }  <-> 
( ( D : M
--> O  /\  C : M
--> O  /\  J : O
--> M )  /\  ( Fun  R  /\  dom  R  C_  ( M  X.  M
)  /\  ran  R  C_  M ) ) ) )
467, 45syl5bb 248 . 2  |-  ( ( ( D  e.  A  /\  C  e.  B  /\  J  e.  F
)  /\  R  e.  G )  ->  ( <. <. D ,  C >. ,  <. J ,  R >. >.  e.  { x  |  E. d E. c E. j E. r ( x  =  <. <. d ,  c >. ,  <. j ,  r >. >.  /\  (
d : dom  d --> dom  j  /\  c : dom  d --> dom  j  /\  j : dom  j --> dom  d )  /\  ( Fun  r  /\  dom  r  C_  ( dom  d  X. 
dom  d )  /\  ran  r  C_  dom  d
) ) }  <->  ( ( D : M --> O  /\  C : M --> O  /\  J : O --> M )  /\  ( Fun  R  /\  dom  R  C_  ( M  X.  M )  /\  ran  R  C_  M )
) ) )
472, 46syl5bb 248 1  |-  ( ( ( D  e.  A  /\  C  e.  B  /\  J  e.  F
)  /\  R  e.  G )  ->  ( <. <. D ,  C >. ,  <. J ,  R >. >.  e.  Alg  <->  ( ( D : M --> O  /\  C : M --> O  /\  J : O --> M )  /\  ( Fun  R  /\  dom  R  C_  ( M  X.  M )  /\  ran  R  C_  M )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    C_ wss 3165   <.cop 3656    X. cxp 4703   dom cdm 4705   ran crn 4706   Fun wfun 5265   -->wf 5267    Alg calg 25814
This theorem is referenced by:  1alg  25825  0alg  25859  dualalg  25885  setiscat  26082
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275  df-alg 25819
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