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Theorem algi 25830
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 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 ) ) )
433exbii 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 ) ) )
54abbii 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
) ) }
65eleq2i 2360 . . 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 25826 . . . . . . 7  |-  D  =  ( 1st `  ( 1st `  T ) )
98eqcomi 2300 . . . . . 6  |-  ( 1st `  ( 1st `  T
) )  =  D
109eqeq2i 2306 . . . . 5  |-  ( d  =  ( 1st `  ( 1st `  T ) )  <-> 
d  =  D )
11 feq1 5391 . . . . . . . 8  |-  ( d  =  D  ->  (
d : dom  d --> dom  j  <->  D : dom  d --> dom  j ) )
12 dmeq 4895 . . . . . . . . . 10  |-  ( d  =  D  ->  dom  d  =  dom  D )
1312feq2d 5396 . . . . . . . . 9  |-  ( d  =  D  ->  ( D : dom  d --> dom  j  <->  D : dom  D --> dom  j ) )
14 algi.5 . . . . . . . . . 10  |-  M  =  dom  D
1514feq2i 5400 . . . . . . . . 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 5392 . . . . . . . . 9  |-  ( dom  d  =  dom  D  ->  ( c : dom  d
--> dom  j  <->  c : dom  D --> dom  j )
)
1914feq2i 5400 . . . . . . . . 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 2346 . . . . . . . 8  |-  ( d  =  D  ->  dom  d  =  M )
23 feq3 5393 . . . . . . . 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 4916 . . . . . . . . 9  |-  ( ( dom  d  X.  dom  d )  =  ( M  X.  M )  <->  dom  d  =  M
)
27 sseq2 3213 . . . . . . . . 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 3213 . . . . . . . 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 25827 . . . . . . 7  |-  C  =  ( 2nd `  ( 1st `  T ) )
3635eqcomi 2300 . . . . . 6  |-  ( 2nd `  ( 1st `  T
) )  =  C
3736eqeq2i 2306 . . . . 5  |-  ( c  =  ( 2nd `  ( 1st `  T ) )  <-> 
c  =  C )
38 feq1 5391 . . . . . . 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 25828 . . . . . . 7  |-  J  =  ( 1st `  ( 2nd `  T ) )
4443eqcomi 2300 . . . . . 6  |-  ( 1st `  ( 2nd `  T
) )  =  J
4544eqeq2i 2306 . . . . 5  |-  ( j  =  ( 1st `  ( 2nd `  T ) )  <-> 
j  =  J )
46 dmeq 4895 . . . . . . . . 9  |-  ( j  =  J  ->  dom  j  =  dom  J )
47 algi.6 . . . . . . . . 9  |-  O  =  dom  J
4846, 47syl6eqr 2346 . . . . . . . 8  |-  ( j  =  J  ->  dom  j  =  O )
49 feq3 5393 . . . . . . . 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 5393 . . . . . . . 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 5391 . . . . . . . 8  |-  ( j  =  J  ->  (
j : dom  j --> M 
<->  J : dom  j --> M ) )
5448feq2d 5396 . . . . . . . 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 25829 . . . . . . 7  |-  R  =  ( 2nd `  ( 2nd `  T ) )
6160eqcomi 2300 . . . . . 6  |-  ( 2nd `  ( 2nd `  T
) )  =  R
6261eqeq2i 2306 . . . . 5  |-  ( r  =  ( 2nd `  ( 2nd `  T ) )  <-> 
r  =  R )
63 funeq 5290 . . . . . . 7  |-  ( r  =  R  ->  ( Fun  r  <->  Fun  R ) )
64 dmeq 4895 . . . . . . . 8  |-  ( r  =  R  ->  dom  r  =  dom  R )
6564sseq1d 3218 . . . . . . 7  |-  ( r  =  R  ->  ( dom  r  C_  ( M  X.  M )  <->  dom  R  C_  ( M  X.  M
) ) )
66 rneq 4920 . . . . . . . 8  |-  ( r  =  R  ->  ran  r  =  ran  R )
6766sseq1d 3218 . . . . . . 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 25189 . . 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 25819 . 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 2388 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 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   ` cfv 5271   1stc1st 6136   2ndc2nd 6137    Alg calg 25814   dom_cdom_ 25815   cod_ccod_ 25816   id_cid_ 25817   o_co_ 25818
This theorem is referenced by:  doma  25831  coda  25832  ida  25833  cmppfa  25835
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528
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-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  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-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139  df-alg 25819  df-dom_ 25820  df-cod_ 25821  df-id_ 25822  df-cmpa 25823
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