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Theorem exidresid 26246
Description: The restriction of a binary operation with identity to a subset containing the identity has the same identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
Hypotheses
Ref Expression
exidres.1  |-  X  =  ran  G
exidres.2  |-  U  =  (GId `  G )
exidres.3  |-  H  =  ( G  |`  ( Y  X.  Y ) )
Assertion
Ref Expression
exidresid  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  (GId `  H )  =  U )

Proof of Theorem exidresid
Dummy variables  x  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exidres.3 . . . . . 6  |-  H  =  ( G  |`  ( Y  X.  Y ) )
2 resexg 5126 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( G  |`  ( Y  X.  Y
) )  e.  _V )
31, 2syl5eqel 2472 . . . . 5  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  H  e.  _V )
4 eqid 2388 . . . . . 6  |-  ran  H  =  ran  H
54gidval 21650 . . . . 5  |-  ( H  e.  _V  ->  (GId `  H )  =  (
iota_ u  e.  ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  ( x H u )  =  x ) ) )
63, 5syl 16 . . . 4  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  (GId `  H
)  =  ( iota_ u  e.  ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  ( x H u )  =  x ) ) )
763ad2ant1 978 . . 3  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  (GId `  H
)  =  ( iota_ u  e.  ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  ( x H u )  =  x ) ) )
87adantr 452 . 2  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  (GId `  H )  =  (
iota_ u  e.  ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  ( x H u )  =  x ) ) )
9 exidres.1 . . . . . . 7  |-  X  =  ran  G
10 exidres.2 . . . . . . 7  |-  U  =  (GId `  G )
119, 10, 1exidreslem 26244 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  ( U  e.  dom  dom  H  /\  A. x  e.  dom  dom  H ( ( U H x )  =  x  /\  ( x H U )  =  x ) ) )
1211simprd 450 . . . . 5  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  A. x  e.  dom  dom  H (
( U H x )  =  x  /\  ( x H U )  =  x ) )
1312adantr 452 . . . 4  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  A. x  e.  dom  dom  H (
( U H x )  =  x  /\  ( x H U )  =  x ) )
149, 10, 1exidres 26245 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  H  e.  ExId 
)
15 elin 3474 . . . . . . . 8  |-  ( H  e.  ( Magma  i^i  ExId  )  <-> 
( H  e.  Magma  /\  H  e.  ExId  )
)
16 rngopid 21760 . . . . . . . 8  |-  ( H  e.  ( Magma  i^i  ExId  )  ->  ran  H  =  dom  dom  H )
1715, 16sylbir 205 . . . . . . 7  |-  ( ( H  e.  Magma  /\  H  e.  ExId  )  ->  ran  H  =  dom  dom  H
)
1817ancoms 440 . . . . . 6  |-  ( ( H  e.  ExId  /\  H  e.  Magma )  ->  ran  H  =  dom  dom  H
)
1914, 18sylan 458 . . . . 5  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  ran  H  =  dom  dom  H
)
2019raleqdv 2854 . . . 4  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  ( A. x  e.  ran  H ( ( U H x )  =  x  /\  ( x H U )  =  x )  <->  A. x  e.  dom  dom 
H ( ( U H x )  =  x  /\  ( x H U )  =  x ) ) )
2113, 20mpbird 224 . . 3  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  A. x  e.  ran  H ( ( U H x )  =  x  /\  (
x H U )  =  x ) )
2211simpld 446 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  U  e.  dom  dom  H )
2322adantr 452 . . . . 5  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  U  e.  dom  dom  H )
2423, 19eleqtrrd 2465 . . . 4  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  U  e.  ran  H )
254exidu1 21763 . . . . . . 7  |-  ( H  e.  ( Magma  i^i  ExId  )  ->  E! u  e. 
ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  (
x H u )  =  x ) )
2615, 25sylbir 205 . . . . . 6  |-  ( ( H  e.  Magma  /\  H  e.  ExId  )  ->  E! u  e.  ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
2726ancoms 440 . . . . 5  |-  ( ( H  e.  ExId  /\  H  e.  Magma )  ->  E! u  e.  ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
2814, 27sylan 458 . . . 4  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  E! u  e.  ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  ( x H u )  =  x ) )
29 oveq1 6028 . . . . . . . 8  |-  ( u  =  U  ->  (
u H x )  =  ( U H x ) )
3029eqeq1d 2396 . . . . . . 7  |-  ( u  =  U  ->  (
( u H x )  =  x  <->  ( U H x )  =  x ) )
31 oveq2 6029 . . . . . . . 8  |-  ( u  =  U  ->  (
x H u )  =  ( x H U ) )
3231eqeq1d 2396 . . . . . . 7  |-  ( u  =  U  ->  (
( x H u )  =  x  <->  ( x H U )  =  x ) )
3330, 32anbi12d 692 . . . . . 6  |-  ( u  =  U  ->  (
( ( u H x )  =  x  /\  ( x H u )  =  x )  <->  ( ( U H x )  =  x  /\  ( x H U )  =  x ) ) )
3433ralbidv 2670 . . . . 5  |-  ( u  =  U  ->  ( A. x  e.  ran  H ( ( u H x )  =  x  /\  ( x H u )  =  x )  <->  A. x  e.  ran  H ( ( U H x )  =  x  /\  ( x H U )  =  x ) ) )
3534riota2 6509 . . . 4  |-  ( ( U  e.  ran  H  /\  E! u  e.  ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  ( x H u )  =  x ) )  ->  ( A. x  e.  ran  H ( ( U H x )  =  x  /\  ( x H U )  =  x )  <->  ( iota_ u  e. 
ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  (
x H u )  =  x ) )  =  U ) )
3624, 28, 35syl2anc 643 . . 3  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  ( A. x  e.  ran  H ( ( U H x )  =  x  /\  ( x H U )  =  x )  <->  ( iota_ u  e. 
ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  (
x H u )  =  x ) )  =  U ) )
3721, 36mpbid 202 . 2  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  ( iota_ u  e.  ran  H A. x  e.  ran  H ( ( u H x )  =  x  /\  ( x H u )  =  x ) )  =  U )
388, 37eqtrd 2420 1  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  (GId `  H )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2650   E!wreu 2652   _Vcvv 2900    i^i cin 3263    C_ wss 3264    X. cxp 4817   dom cdm 4819   ran crn 4820    |` cres 4821   ` cfv 5395  (class class class)co 6021   iota_crio 6479  GIdcgi 21624    ExId cexid 21751   Magmacmagm 21755
This theorem is referenced by:  isdrngo2  26266
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-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  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-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fo 5401  df-fv 5403  df-ov 6024  df-riota 6486  df-gid 21629  df-exid 21752  df-mgm 21756
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