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Theorem exidresid 26545
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 5177 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( G  |`  ( Y  X.  Y
) )  e.  _V )
31, 2syl5eqel 2519 . . . . 5  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  H  e.  _V )
4 eqid 2435 . . . . . 6  |-  ran  H  =  ran  H
54gidval 21793 . . . . 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 26543 . . . . . 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 26544 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  H  e.  ExId 
)
15 elin 3522 . . . . . . . 8  |-  ( H  e.  ( Magma  i^i  ExId  )  <-> 
( H  e.  Magma  /\  H  e.  ExId  )
)
16 rngopid 21903 . . . . . . . 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 2902 . . . 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 2512 . . . 4  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  U  e.  ran  H )
254exidu1 21906 . . . . . . 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 6080 . . . . . . . 8  |-  ( u  =  U  ->  (
u H x )  =  ( U H x ) )
3029eqeq1d 2443 . . . . . . 7  |-  ( u  =  U  ->  (
( u H x )  =  x  <->  ( U H x )  =  x ) )
31 oveq2 6081 . . . . . . . 8  |-  ( u  =  U  ->  (
x H u )  =  ( x H U ) )
3231eqeq1d 2443 . . . . . . 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 2717 . . . . 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 6564 . . . 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 2467 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 1652    e. wcel 1725   A.wral 2697   E!wreu 2699   _Vcvv 2948    i^i cin 3311    C_ wss 3312    X. cxp 4868   dom cdm 4870   ran crn 4871    |` cres 4872   ` cfv 5446  (class class class)co 6073   iota_crio 6534  GIdcgi 21767    ExId cexid 21894   Magmacmagm 21898
This theorem is referenced by:  isdrngo2  26565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-riota 6541  df-gid 21772  df-exid 21895  df-mgm 21899
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