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Theorem algi 25727
Description: "Axiomatic" properties of  Alg. (Contributed by FL, 24-Oct-2007.)
Hypotheses
Ref Expression
algi.1  |-  D  =  ( dom_ `  T
)
algi.2  |-  C  =  ( cod_ `  T
)
algi.3  |-  J  =  ( id_ `  T
)
algi.4  |-  R  =  ( o_ `  T
)
algi.5  |-  M  =  dom  D
algi.6  |-  O  =  dom  J
Assertion
Ref Expression
algi  |-  ( T  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 algi
Dummy variables  c 
d  j  r  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3anass 938 . . . . . . . 8  |-  ( ( 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
) ) ) )
21bicomi 193 . . . . . . 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 ) ) )
32exbii 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 ) ) )
433exbii 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 ) ) )
54abbii 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
) ) }
65eleq2i 2347 . . 3  |-  ( T  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
) ) ) }  <-> 
T  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
) ) } )
7 algi.1 . . . . . . . 8  |-  D  =  ( dom_ `  T
)
87domval 25723 . . . . . . 7  |-  D  =  ( 1st `  ( 1st `  T ) )
98eqcomi 2287 . . . . . 6  |-  ( 1st `  ( 1st `  T
) )  =  D
109eqeq2i 2293 . . . . 5  |-  ( d  =  ( 1st `  ( 1st `  T ) )  <-> 
d  =  D )
11 feq1 5375 . . . . . . . 8  |-  ( d  =  D  ->  (
d : dom  d --> dom  j  <->  D : dom  d --> dom  j ) )
12 dmeq 4879 . . . . . . . . . 10  |-  ( d  =  D  ->  dom  d  =  dom  D )
1312feq2d 5380 . . . . . . . . 9  |-  ( d  =  D  ->  ( D : dom  d --> dom  j  <->  D : dom  D --> dom  j ) )
14 algi.5 . . . . . . . . . 10  |-  M  =  dom  D
1514feq2i 5384 . . . . . . . . 9  |-  ( D : M --> dom  j  <->  D : dom  D --> dom  j
)
1613, 15syl6bbr 254 . . . . . . . 8  |-  ( d  =  D  ->  ( D : dom  d --> dom  j  <->  D : M --> dom  j
) )
1711, 16bitrd 244 . . . . . . 7  |-  ( d  =  D  ->  (
d : dom  d --> dom  j  <->  D : M --> dom  j
) )
18 feq2 5376 . . . . . . . . 9  |-  ( dom  d  =  dom  D  ->  ( c : dom  d
--> dom  j  <->  c : dom  D --> dom  j )
)
1914feq2i 5384 . . . . . . . . 9  |-  ( c : M --> dom  j  <->  c : dom  D --> dom  j
)
2018, 19syl6bbr 254 . . . . . . . 8  |-  ( dom  d  =  dom  D  ->  ( c : dom  d
--> dom  j  <->  c : M
--> dom  j ) )
2112, 20syl 15 . . . . . . 7  |-  ( d  =  D  ->  (
c : dom  d --> dom  j  <->  c : M --> dom  j ) )
2212, 14syl6eqr 2333 . . . . . . . 8  |-  ( d  =  D  ->  dom  d  =  M )
23 feq3 5377 . . . . . . . 8  |-  ( dom  d  =  M  -> 
( j : dom  j
--> dom  d  <->  j : dom  j --> M ) )
2422, 23syl 15 . . . . . . 7  |-  ( d  =  D  ->  (
j : dom  j --> dom  d  <->  j : dom  j
--> M ) )
2517, 21, 243anbi123d 1252 . . . . . 6  |-  ( 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 ) ) )
26 xpid11 4900 . . . . . . . . 9  |-  ( ( dom  d  X.  dom  d )  =  ( M  X.  M )  <->  dom  d  =  M
)
27 sseq2 3200 . . . . . . . . 9  |-  ( ( dom  d  X.  dom  d )  =  ( M  X.  M )  ->  ( dom  r  C_  ( dom  d  X. 
dom  d )  <->  dom  r  C_  ( M  X.  M
) ) )
2826, 27sylbir 204 . . . . . . . 8  |-  ( dom  d  =  M  -> 
( dom  r  C_  ( dom  d  X.  dom  d )  <->  dom  r  C_  ( M  X.  M
) ) )
29 sseq2 3200 . . . . . . . 8  |-  ( dom  d  =  M  -> 
( ran  r  C_  dom  d  <->  ran  r  C_  M
) )
3028, 293anbi23d 1255 . . . . . . 7  |-  ( dom  d  =  M  -> 
( ( 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 ) ) )
3122, 30syl 15 . . . . . 6  |-  ( 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 ) ) )
3225, 31anbi12d 691 . . . . 5  |-  ( 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 ) ) ) )
3310, 32sylbi 187 . . . 4  |-  ( d  =  ( 1st `  ( 1st `  T ) )  ->  ( ( ( 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 ) ) ) )
34 algi.2 . . . . . . . 8  |-  C  =  ( cod_ `  T
)
3534codval 25724 . . . . . . 7  |-  C  =  ( 2nd `  ( 1st `  T ) )
3635eqcomi 2287 . . . . . 6  |-  ( 2nd `  ( 1st `  T
) )  =  C
3736eqeq2i 2293 . . . . 5  |-  ( c  =  ( 2nd `  ( 1st `  T ) )  <-> 
c  =  C )
38 feq1 5375 . . . . . . 7  |-  ( c  =  C  ->  (
c : M --> dom  j  <->  C : M --> dom  j
) )
39383anbi2d 1257 . . . . . 6  |-  ( 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 ) ) )
4039anbi1d 685 . . . . 5  |-  ( 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 ) ) ) )
4137, 40sylbi 187 . . . 4  |-  ( c  =  ( 2nd `  ( 1st `  T ) )  ->  ( ( ( 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 ) ) ) )
42 algi.3 . . . . . . . 8  |-  J  =  ( id_ `  T
)
4342idval 25725 . . . . . . 7  |-  J  =  ( 1st `  ( 2nd `  T ) )
4443eqcomi 2287 . . . . . 6  |-  ( 1st `  ( 2nd `  T
) )  =  J
4544eqeq2i 2293 . . . . 5  |-  ( j  =  ( 1st `  ( 2nd `  T ) )  <-> 
j  =  J )
46 dmeq 4879 . . . . . . . . 9  |-  ( j  =  J  ->  dom  j  =  dom  J )
47 algi.6 . . . . . . . . 9  |-  O  =  dom  J
4846, 47syl6eqr 2333 . . . . . . . 8  |-  ( j  =  J  ->  dom  j  =  O )
49 feq3 5377 . . . . . . . 8  |-  ( dom  j  =  O  -> 
( D : M --> dom  j  <->  D : M --> O ) )
5048, 49syl 15 . . . . . . 7  |-  ( j  =  J  ->  ( D : M --> dom  j  <->  D : M --> O ) )
51 feq3 5377 . . . . . . . 8  |-  ( dom  j  =  O  -> 
( C : M --> dom  j  <->  C : M --> O ) )
5248, 51syl 15 . . . . . . 7  |-  ( j  =  J  ->  ( C : M --> dom  j  <->  C : M --> O ) )
53 feq1 5375 . . . . . . . 8  |-  ( j  =  J  ->  (
j : dom  j --> M 
<->  J : dom  j --> M ) )
5448feq2d 5380 . . . . . . . 8  |-  ( j  =  J  ->  ( J : dom  j --> M  <-> 
J : O --> M ) )
5553, 54bitrd 244 . . . . . . 7  |-  ( j  =  J  ->  (
j : dom  j --> M 
<->  J : O --> M ) )
5650, 52, 553anbi123d 1252 . . . . . 6  |-  ( j  =  J  ->  (
( D : M --> dom  j  /\  C : M
--> dom  j  /\  j : dom  j --> M )  <-> 
( D : M --> O  /\  C : M --> O  /\  J : O --> M ) ) )
5756anbi1d 685 . . . . 5  |-  ( 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 ) ) ) )
5845, 57sylbi 187 . . . 4  |-  ( j  =  ( 1st `  ( 2nd `  T ) )  ->  ( ( ( 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 ) ) ) )
59 algi.4 . . . . . . . 8  |-  R  =  ( o_ `  T
)
6059cmpval 25726 . . . . . . 7  |-  R  =  ( 2nd `  ( 2nd `  T ) )
6160eqcomi 2287 . . . . . 6  |-  ( 2nd `  ( 2nd `  T
) )  =  R
6261eqeq2i 2293 . . . . 5  |-  ( r  =  ( 2nd `  ( 2nd `  T ) )  <-> 
r  =  R )
63 funeq 5274 . . . . . . 7  |-  ( r  =  R  ->  ( Fun  r  <->  Fun  R ) )
64 dmeq 4879 . . . . . . . 8  |-  ( r  =  R  ->  dom  r  =  dom  R )
6564sseq1d 3205 . . . . . . 7  |-  ( r  =  R  ->  ( dom  r  C_  ( M  X.  M )  <->  dom  R  C_  ( M  X.  M
) ) )
66 rneq 4904 . . . . . . . 8  |-  ( r  =  R  ->  ran  r  =  ran  R )
6766sseq1d 3205 . . . . . . 7  |-  ( r  =  R  ->  ( ran  r  C_  M  <->  ran  R  C_  M ) )
6863, 65, 673anbi123d 1252 . . . . . 6  |-  ( 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 ) ) )
6968anbi2d 684 . . . . 5  |-  ( 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 )
) ) )
7062, 69sylbi 187 . . . 4  |-  ( r  =  ( 2nd `  ( 2nd `  T ) )  ->  ( ( ( 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 )
) ) )
7133, 41, 58, 70eloi 25086 . . 3  |-  ( T  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 )
) )
726, 71sylbir 204 . 2  |-  ( T  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 ) ) )
73 df-alg 25716 . 2  |-  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 ) ) }
7472, 73eleq2s 2375 1  |-  ( T  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   ` cfv 5255   1stc1st 6120   2ndc2nd 6121    Alg calg 25711   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714   o_co_ 25715
This theorem is referenced by:  doma  25728  coda  25729  ida  25730  cmppfa  25732
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-alg 25716  df-dom_ 25717  df-cod_ 25718  df-id_ 25719  df-cmpa 25720
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