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Theorem exidresid 29795
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 5307 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( G  |`  ( Y  X.  Y
) )  e.  _V )
31, 2syl5eqel 2552 . . . . 5  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  H  e.  _V )
4 eqid 2460 . . . . . 6  |-  ran  H  =  ran  H
54gidval 24741 . . . . 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 1012 . . 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 465 . 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 29793 . . . . . 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 463 . . . . 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 465 . . . 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 29794 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  H  e.  ExId 
)
15 elin 3680 . . . . . . . 8  |-  ( H  e.  ( Magma  i^i  ExId  )  <-> 
( H  e.  Magma  /\  H  e.  ExId  )
)
16 rngopid 24851 . . . . . . . 8  |-  ( H  e.  ( Magma  i^i  ExId  )  ->  ran  H  =  dom  dom  H )
1715, 16sylbir 213 . . . . . . 7  |-  ( ( H  e.  Magma  /\  H  e.  ExId  )  ->  ran  H  =  dom  dom  H
)
1817ancoms 453 . . . . . 6  |-  ( ( H  e.  ExId  /\  H  e.  Magma )  ->  ran  H  =  dom  dom  H
)
1914, 18sylan 471 . . . . 5  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  ran  H  =  dom  dom  H
)
2019raleqdv 3057 . . . 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 232 . . 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 459 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  U  e.  dom  dom  H )
2322adantr 465 . . . . 5  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  U  e.  dom  dom  H )
2423, 19eleqtrrd 2551 . . . 4  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  U  e.  ran  H )
254exidu1 24854 . . . . . . 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 213 . . . . . 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 453 . . . . 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 471 . . . 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 6282 . . . . . . . 8  |-  ( u  =  U  ->  (
u H x )  =  ( U H x ) )
3029eqeq1d 2462 . . . . . . 7  |-  ( u  =  U  ->  (
( u H x )  =  x  <->  ( U H x )  =  x ) )
31 oveq2 6283 . . . . . . . 8  |-  ( u  =  U  ->  (
x H u )  =  ( x H U ) )
3231eqeq1d 2462 . . . . . . 7  |-  ( u  =  U  ->  (
( x H u )  =  x  <->  ( x H U )  =  x ) )
3330, 32anbi12d 710 . . . . . 6  |-  ( u  =  U  ->  (
( ( u H x )  =  x  /\  ( x H u )  =  x )  <->  ( ( U H x )  =  x  /\  ( x H U )  =  x ) ) )
3433ralbidv 2896 . . . . 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 6259 . . . 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 661 . . 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 210 . 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 2501 1  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  (GId `  H )  =  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   E!wreu 2809   _Vcvv 3106    i^i cin 3468    C_ wss 3469    X. cxp 4990   dom cdm 4992   ran crn 4993    |` cres 4994   ` cfv 5579   iota_crio 6235  (class class class)co 6275  GIdcgi 24715    ExId cexid 24842   Magmacmagm 24846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fo 5585  df-fv 5587  df-riota 6236  df-ov 6278  df-gid 24720  df-exid 24843  df-mgm 24847
This theorem is referenced by:  isdrngo2  29815
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