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Theorem isalg 25721
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 25716 . . 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 2347 . 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 1569 . . . . . 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 1571 . . . . 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 2395 . . . 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 2347 . . 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 5375 . . . . . . 7  |-  ( d  =  D  ->  (
d : dom  d --> dom  j  <->  D : dom  d --> dom  j ) )
9 dmeq 4879 . . . . . . . . 9  |-  ( d  =  D  ->  dom  d  =  dom  D )
10 isalg.1 . . . . . . . . 9  |-  M  =  dom  D
119, 10syl6eqr 2333 . . . . . . . 8  |-  ( d  =  D  ->  dom  d  =  M )
1211feq2d 5380 . . . . . . 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 5380 . . . . . 6  |-  ( d  =  D  ->  (
c : dom  d --> dom  j  <->  c : M --> dom  j ) )
15 feq3 5377 . . . . . . 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 4714 . . . . . . 7  |-  ( d  =  D  ->  ( dom  d  X.  dom  d
)  =  ( M  X.  M ) )
1918sseq2d 3206 . . . . . 6  |-  ( d  =  D  ->  ( dom  r  C_  ( dom  d  X.  dom  d
)  <->  dom  r  C_  ( M  X.  M ) ) )
2011sseq2d 3206 . . . . . 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 5375 . . . . . 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 4879 . . . . . . . 8  |-  ( j  =  J  ->  dom  j  =  dom  J )
27 isalg.2 . . . . . . . 8  |-  O  =  dom  J
2826, 27syl6eqr 2333 . . . . . . 7  |-  ( j  =  J  ->  dom  j  =  O )
29 feq3 5377 . . . . . . 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 5377 . . . . . . 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 5375 . . . . . . 7  |-  ( j  =  J  ->  (
j : dom  j --> M 
<->  J : dom  j --> M ) )
3428feq2d 5380 . . . . . . 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 5274 . . . . . 6  |-  ( r  =  R  ->  ( Fun  r  <->  Fun  R ) )
39 dmeq 4879 . . . . . . 7  |-  ( r  =  R  ->  dom  r  =  dom  R )
4039sseq1d 3205 . . . . . 6  |-  ( r  =  R  ->  ( dom  r  C_  ( M  X.  M )  <->  dom  R  C_  ( M  X.  M
) ) )
41 rneq 4904 . . . . . . 7  |-  ( r  =  R  ->  ran  r  =  ran  R )
4241sseq1d 3205 . . . . . 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 25041 . . 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 1528    = wceq 1623    e. wcel 1684   {cab 2269    C_ wss 3152   <.cop 3643    X. cxp 4687   dom cdm 4689   ran crn 4690   Fun wfun 5249   -->wf 5251    Alg calg 25711
This theorem is referenced by:  1alg  25722  0alg  25756  dualalg  25782  setiscat  25979
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259  df-alg 25716
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