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Theorem exidresid 28879
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 5244 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( G  |`  ( Y  X.  Y
) )  e.  _V )
31, 2syl5eqel 2541 . . . . 5  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  H  e.  _V )
4 eqid 2451 . . . . . 6  |-  ran  H  =  ran  H
54gidval 23832 . . . . 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 1009 . . 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 28877 . . . . . 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 28878 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  H  e.  ExId 
)
15 elin 3634 . . . . . . . 8  |-  ( H  e.  ( Magma  i^i  ExId  )  <-> 
( H  e.  Magma  /\  H  e.  ExId  )
)
16 rngopid 23942 . . . . . . . 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 3016 . . . 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 2540 . . . 4  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  U  e.  ran  H )
254exidu1 23945 . . . . . . 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 6194 . . . . . . . 8  |-  ( u  =  U  ->  (
u H x )  =  ( U H x ) )
3029eqeq1d 2453 . . . . . . 7  |-  ( u  =  U  ->  (
( u H x )  =  x  <->  ( U H x )  =  x ) )
31 oveq2 6195 . . . . . . . 8  |-  ( u  =  U  ->  (
x H u )  =  ( x H U ) )
3231eqeq1d 2453 . . . . . . 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 2839 . . . . 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 6171 . . . 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 2491 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 965    = wceq 1370    e. wcel 1758   A.wral 2793   E!wreu 2795   _Vcvv 3065    i^i cin 3422    C_ wss 3423    X. cxp 4933   dom cdm 4935   ran crn 4936    |` cres 4937   ` cfv 5513   iota_crio 6147  (class class class)co 6187  GIdcgi 23806    ExId cexid 23933   Magmacmagm 23937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-fo 5519  df-fv 5521  df-riota 6148  df-ov 6190  df-gid 23811  df-exid 23934  df-mgm 23938
This theorem is referenced by:  isdrngo2  28899
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