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Theorem ismgm 20987
Description: The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
Hypothesis
Ref Expression
ismgm.1  |-  X  =  dom  dom  G
Assertion
Ref Expression
ismgm  |-  ( G  e.  A  ->  ( G  e.  Magma  <->  G :
( X  X.  X
) --> X ) )

Proof of Theorem ismgm
Dummy variables  g 
t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 feq1 5375 . . . . 5  |-  ( g  =  G  ->  (
g : ( t  X.  t ) --> t  <-> 
G : ( t  X.  t ) --> t ) )
21exbidv 1612 . . . 4  |-  ( g  =  G  ->  ( E. t  g :
( t  X.  t
) --> t  <->  E. t  G : ( t  X.  t ) --> t ) )
3 df-mgm 20986 . . . 4  |-  Magma  =  {
g  |  E. t 
g : ( t  X.  t ) --> t }
42, 3elab2g 2916 . . 3  |-  ( G  e.  A  ->  ( G  e.  Magma  <->  E. t  G : ( t  X.  t ) --> t ) )
5 f00 5426 . . . . . . . 8  |-  ( G : ( (/)  X.  (/) ) --> (/)  <->  ( G  =  (/)  /\  ( (/) 
X.  (/) )  =  (/) ) )
6 dmeq 4879 . . . . . . . . . 10  |-  ( G  =  (/)  ->  dom  G  =  dom  (/) )
7 dmeq 4879 . . . . . . . . . . 11  |-  ( dom 
G  =  dom  (/)  ->  dom  dom 
G  =  dom  dom  (/) )
8 dm0 4892 . . . . . . . . . . . . 13  |-  dom  (/)  =  (/)
98dmeqi 4880 . . . . . . . . . . . 12  |-  dom  dom  (/)  =  dom  (/)
109, 8eqtri 2303 . . . . . . . . . . 11  |-  dom  dom  (/)  =  (/)
117, 10syl6req 2332 . . . . . . . . . 10  |-  ( dom 
G  =  dom  (/)  ->  (/)  =  dom  dom 
G )
126, 11syl 15 . . . . . . . . 9  |-  ( G  =  (/)  ->  (/)  =  dom  dom 
G )
1312adantr 451 . . . . . . . 8  |-  ( ( G  =  (/)  /\  ( (/) 
X.  (/) )  =  (/) )  ->  (/)  =  dom  dom  G )
145, 13sylbi 187 . . . . . . 7  |-  ( G : ( (/)  X.  (/) ) --> (/)  -> 
(/)  =  dom  dom  G )
15 xpeq12 4708 . . . . . . . . . 10  |-  ( ( t  =  (/)  /\  t  =  (/) )  ->  (
t  X.  t )  =  ( (/)  X.  (/) ) )
1615anidms 626 . . . . . . . . 9  |-  ( t  =  (/)  ->  ( t  X.  t )  =  ( (/)  X.  (/) ) )
17 feq23 5378 . . . . . . . . 9  |-  ( ( ( t  X.  t
)  =  ( (/)  X.  (/) )  /\  t  =  (/) )  ->  ( G : ( t  X.  t ) --> t  <->  G :
( (/)  X.  (/) ) --> (/) ) )
1816, 17mpancom 650 . . . . . . . 8  |-  ( t  =  (/)  ->  ( G : ( t  X.  t ) --> t  <->  G :
( (/)  X.  (/) ) --> (/) ) )
19 eqeq1 2289 . . . . . . . 8  |-  ( t  =  (/)  ->  ( t  =  dom  dom  G  <->  (/)  =  dom  dom  G )
)
2018, 19imbi12d 311 . . . . . . 7  |-  ( t  =  (/)  ->  ( ( G : ( t  X.  t ) --> t  ->  t  =  dom  dom 
G )  <->  ( G : ( (/)  X.  (/) ) --> (/)  -> 
(/)  =  dom  dom  G ) ) )
2114, 20mpbiri 224 . . . . . 6  |-  ( t  =  (/)  ->  ( G : ( t  X.  t ) --> t  -> 
t  =  dom  dom  G ) )
22 fdm 5393 . . . . . . . 8  |-  ( G : ( t  X.  t ) --> t  ->  dom  G  =  ( t  X.  t ) )
23 dmeq 4879 . . . . . . . 8  |-  ( dom 
G  =  ( t  X.  t )  ->  dom  dom  G  =  dom  ( t  X.  t
) )
24 df-ne 2448 . . . . . . . . . . . 12  |-  ( t  =/=  (/)  <->  -.  t  =  (/) )
25 dmxp 4897 . . . . . . . . . . . 12  |-  ( t  =/=  (/)  ->  dom  ( t  X.  t )  =  t )
2624, 25sylbir 204 . . . . . . . . . . 11  |-  ( -.  t  =  (/)  ->  dom  ( t  X.  t
)  =  t )
2726eqeq1d 2291 . . . . . . . . . 10  |-  ( -.  t  =  (/)  ->  ( dom  ( t  X.  t
)  =  dom  dom  G  <-> 
t  =  dom  dom  G ) )
2827biimpcd 215 . . . . . . . . 9  |-  ( dom  ( t  X.  t
)  =  dom  dom  G  ->  ( -.  t  =  (/)  ->  t  =  dom  dom  G ) )
2928eqcoms 2286 . . . . . . . 8  |-  ( dom 
dom  G  =  dom  ( t  X.  t
)  ->  ( -.  t  =  (/)  ->  t  =  dom  dom  G )
)
3022, 23, 293syl 18 . . . . . . 7  |-  ( G : ( t  X.  t ) --> t  -> 
( -.  t  =  (/)  ->  t  =  dom  dom 
G ) )
3130com12 27 . . . . . 6  |-  ( -.  t  =  (/)  ->  ( G : ( t  X.  t ) --> t  -> 
t  =  dom  dom  G ) )
3221, 31pm2.61i 156 . . . . 5  |-  ( G : ( t  X.  t ) --> t  -> 
t  =  dom  dom  G )
3332pm4.71ri 614 . . . 4  |-  ( G : ( t  X.  t ) --> t  <->  ( t  =  dom  dom  G  /\  G : ( t  X.  t ) --> t ) )
3433exbii 1569 . . 3  |-  ( E. t  G : ( t  X.  t ) --> t  <->  E. t ( t  =  dom  dom  G  /\  G : ( t  X.  t ) --> t ) )
354, 34syl6bb 252 . 2  |-  ( G  e.  A  ->  ( G  e.  Magma  <->  E. t
( t  =  dom  dom 
G  /\  G :
( t  X.  t
) --> t ) ) )
36 dmexg 4939 . . 3  |-  ( G  e.  A  ->  dom  G  e.  _V )
37 dmexg 4939 . . 3  |-  ( dom 
G  e.  _V  ->  dom 
dom  G  e.  _V )
38 xpeq12 4708 . . . . . . 7  |-  ( ( t  =  dom  dom  G  /\  t  =  dom  dom 
G )  ->  (
t  X.  t )  =  ( dom  dom  G  X.  dom  dom  G
) )
3938anidms 626 . . . . . 6  |-  ( t  =  dom  dom  G  ->  ( t  X.  t
)  =  ( dom 
dom  G  X.  dom  dom  G ) )
40 feq23 5378 . . . . . 6  |-  ( ( ( t  X.  t
)  =  ( dom 
dom  G  X.  dom  dom  G )  /\  t  =  dom  dom  G )  ->  ( G : ( t  X.  t ) --> t  <->  G : ( dom 
dom  G  X.  dom  dom  G ) --> dom  dom  G ) )
4139, 40mpancom 650 . . . . 5  |-  ( t  =  dom  dom  G  ->  ( G : ( t  X.  t ) --> t  <->  G : ( dom 
dom  G  X.  dom  dom  G ) --> dom  dom  G ) )
42 ismgm.1 . . . . . . . 8  |-  X  =  dom  dom  G
4342eqcomi 2287 . . . . . . 7  |-  dom  dom  G  =  X
4443, 43xpeq12i 4711 . . . . . 6  |-  ( dom 
dom  G  X.  dom  dom  G )  =  ( X  X.  X )
4544, 43feq23i 5385 . . . . 5  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) --> dom  dom  G  <->  G :
( X  X.  X
) --> X )
4641, 45syl6bb 252 . . . 4  |-  ( t  =  dom  dom  G  ->  ( G : ( t  X.  t ) --> t  <->  G : ( X  X.  X ) --> X ) )
4746ceqsexgv 2900 . . 3  |-  ( dom 
dom  G  e.  _V  ->  ( E. t ( t  =  dom  dom  G  /\  G : ( t  X.  t ) --> t )  <->  G :
( X  X.  X
) --> X ) )
4836, 37, 473syl 18 . 2  |-  ( G  e.  A  ->  ( E. t ( t  =  dom  dom  G  /\  G : ( t  X.  t ) --> t )  <-> 
G : ( X  X.  X ) --> X ) )
4935, 48bitrd 244 1  |-  ( G  e.  A  ->  ( G  e.  Magma  <->  G :
( X  X.  X
) --> X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455    X. cxp 4687   dom cdm 4689   -->wf 5251   Magmacmagm 20985
This theorem is referenced by:  clmgm  20988  opidon  20989  issmgrp  21001  mgmlion  25337  mgmrddd  25366
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-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-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-id 4309  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-mgm 20986
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