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Theorem ismgm 21757
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 5517 . . . . 5  |-  ( g  =  G  ->  (
g : ( t  X.  t ) --> t  <-> 
G : ( t  X.  t ) --> t ) )
21exbidv 1633 . . . 4  |-  ( g  =  G  ->  ( E. t  g :
( t  X.  t
) --> t  <->  E. t  G : ( t  X.  t ) --> t ) )
3 df-mgm 21756 . . . 4  |-  Magma  =  {
g  |  E. t 
g : ( t  X.  t ) --> t }
42, 3elab2g 3028 . . 3  |-  ( G  e.  A  ->  ( G  e.  Magma  <->  E. t  G : ( t  X.  t ) --> t ) )
5 f00 5569 . . . . . . . 8  |-  ( G : ( (/)  X.  (/) ) --> (/)  <->  ( G  =  (/)  /\  ( (/) 
X.  (/) )  =  (/) ) )
6 dmeq 5011 . . . . . . . . . 10  |-  ( G  =  (/)  ->  dom  G  =  dom  (/) )
7 dmeq 5011 . . . . . . . . . . 11  |-  ( dom 
G  =  dom  (/)  ->  dom  dom 
G  =  dom  dom  (/) )
8 dm0 5024 . . . . . . . . . . . . 13  |-  dom  (/)  =  (/)
98dmeqi 5012 . . . . . . . . . . . 12  |-  dom  dom  (/)  =  dom  (/)
109, 8eqtri 2408 . . . . . . . . . . 11  |-  dom  dom  (/)  =  (/)
117, 10syl6req 2437 . . . . . . . . . 10  |-  ( dom 
G  =  dom  (/)  ->  (/)  =  dom  dom 
G )
126, 11syl 16 . . . . . . . . 9  |-  ( G  =  (/)  ->  (/)  =  dom  dom 
G )
1312adantr 452 . . . . . . . 8  |-  ( ( G  =  (/)  /\  ( (/) 
X.  (/) )  =  (/) )  ->  (/)  =  dom  dom  G )
145, 13sylbi 188 . . . . . . 7  |-  ( G : ( (/)  X.  (/) ) --> (/)  -> 
(/)  =  dom  dom  G )
15 xpeq12 4838 . . . . . . . . . 10  |-  ( ( t  =  (/)  /\  t  =  (/) )  ->  (
t  X.  t )  =  ( (/)  X.  (/) ) )
1615anidms 627 . . . . . . . . 9  |-  ( t  =  (/)  ->  ( t  X.  t )  =  ( (/)  X.  (/) ) )
17 feq23 5520 . . . . . . . . 9  |-  ( ( ( t  X.  t
)  =  ( (/)  X.  (/) )  /\  t  =  (/) )  ->  ( G : ( t  X.  t ) --> t  <->  G :
( (/)  X.  (/) ) --> (/) ) )
1816, 17mpancom 651 . . . . . . . 8  |-  ( t  =  (/)  ->  ( G : ( t  X.  t ) --> t  <->  G :
( (/)  X.  (/) ) --> (/) ) )
19 eqeq1 2394 . . . . . . . 8  |-  ( t  =  (/)  ->  ( t  =  dom  dom  G  <->  (/)  =  dom  dom  G )
)
2018, 19imbi12d 312 . . . . . . 7  |-  ( t  =  (/)  ->  ( ( G : ( t  X.  t ) --> t  ->  t  =  dom  dom 
G )  <->  ( G : ( (/)  X.  (/) ) --> (/)  -> 
(/)  =  dom  dom  G ) ) )
2114, 20mpbiri 225 . . . . . 6  |-  ( t  =  (/)  ->  ( G : ( t  X.  t ) --> t  -> 
t  =  dom  dom  G ) )
22 fdm 5536 . . . . . . . 8  |-  ( G : ( t  X.  t ) --> t  ->  dom  G  =  ( t  X.  t ) )
23 dmeq 5011 . . . . . . . 8  |-  ( dom 
G  =  ( t  X.  t )  ->  dom  dom  G  =  dom  ( t  X.  t
) )
24 df-ne 2553 . . . . . . . . . . . 12  |-  ( t  =/=  (/)  <->  -.  t  =  (/) )
25 dmxp 5029 . . . . . . . . . . . 12  |-  ( t  =/=  (/)  ->  dom  ( t  X.  t )  =  t )
2624, 25sylbir 205 . . . . . . . . . . 11  |-  ( -.  t  =  (/)  ->  dom  ( t  X.  t
)  =  t )
2726eqeq1d 2396 . . . . . . . . . 10  |-  ( -.  t  =  (/)  ->  ( dom  ( t  X.  t
)  =  dom  dom  G  <-> 
t  =  dom  dom  G ) )
2827biimpcd 216 . . . . . . . . 9  |-  ( dom  ( t  X.  t
)  =  dom  dom  G  ->  ( -.  t  =  (/)  ->  t  =  dom  dom  G ) )
2928eqcoms 2391 . . . . . . . 8  |-  ( dom 
dom  G  =  dom  ( t  X.  t
)  ->  ( -.  t  =  (/)  ->  t  =  dom  dom  G )
)
3022, 23, 293syl 19 . . . . . . 7  |-  ( G : ( t  X.  t ) --> t  -> 
( -.  t  =  (/)  ->  t  =  dom  dom 
G ) )
3130com12 29 . . . . . 6  |-  ( -.  t  =  (/)  ->  ( G : ( t  X.  t ) --> t  -> 
t  =  dom  dom  G ) )
3221, 31pm2.61i 158 . . . . 5  |-  ( G : ( t  X.  t ) --> t  -> 
t  =  dom  dom  G )
3332pm4.71ri 615 . . . 4  |-  ( G : ( t  X.  t ) --> t  <->  ( t  =  dom  dom  G  /\  G : ( t  X.  t ) --> t ) )
3433exbii 1589 . . 3  |-  ( E. t  G : ( t  X.  t ) --> t  <->  E. t ( t  =  dom  dom  G  /\  G : ( t  X.  t ) --> t ) )
354, 34syl6bb 253 . 2  |-  ( G  e.  A  ->  ( G  e.  Magma  <->  E. t
( t  =  dom  dom 
G  /\  G :
( t  X.  t
) --> t ) ) )
36 dmexg 5071 . . 3  |-  ( G  e.  A  ->  dom  G  e.  _V )
37 dmexg 5071 . . 3  |-  ( dom 
G  e.  _V  ->  dom 
dom  G  e.  _V )
38 xpeq12 4838 . . . . . . 7  |-  ( ( t  =  dom  dom  G  /\  t  =  dom  dom 
G )  ->  (
t  X.  t )  =  ( dom  dom  G  X.  dom  dom  G
) )
3938anidms 627 . . . . . 6  |-  ( t  =  dom  dom  G  ->  ( t  X.  t
)  =  ( dom 
dom  G  X.  dom  dom  G ) )
40 feq23 5520 . . . . . 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 651 . . . . 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 2392 . . . . . . 7  |-  dom  dom  G  =  X
4443, 43xpeq12i 4841 . . . . . 6  |-  ( dom 
dom  G  X.  dom  dom  G )  =  ( X  X.  X )
4544, 43feq23i 5528 . . . . 5  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) --> dom  dom  G  <->  G :
( X  X.  X
) --> X )
4641, 45syl6bb 253 . . . 4  |-  ( t  =  dom  dom  G  ->  ( G : ( t  X.  t ) --> t  <->  G : ( X  X.  X ) --> X ) )
4746ceqsexgv 3012 . . 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 19 . 2  |-  ( G  e.  A  ->  ( E. t ( t  =  dom  dom  G  /\  G : ( t  X.  t ) --> t )  <-> 
G : ( X  X.  X ) --> X ) )
4935, 48bitrd 245 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 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2551   _Vcvv 2900   (/)c0 3572    X. cxp 4817   dom cdm 4819   -->wf 5391   Magmacmagm 21755
This theorem is referenced by:  clmgm  21758  opidon  21759  issmgrp  21771
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-fun 5397  df-fn 5398  df-f 5399  df-mgm 21756
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