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Theorem exidresid 31623
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 5136 . . . . . 6  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( G  |`  ( Y  X.  Y
) )  e.  _V )
31, 2syl5eqel 2494 . . . . 5  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  H  e.  _V )
4 eqid 2402 . . . . . 6  |-  ran  H  =  ran  H
54gidval 25629 . . . . 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 17 . . . 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 1018 . . 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 463 . 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 31621 . . . . . 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 461 . . . . 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 463 . . . 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 31622 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  H  e.  ExId 
)
15 elin 3626 . . . . . . . 8  |-  ( H  e.  ( Magma  i^i  ExId  )  <-> 
( H  e.  Magma  /\  H  e.  ExId  )
)
16 rngopidOLD 25739 . . . . . . . 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 451 . . . . . 6  |-  ( ( H  e.  ExId  /\  H  e.  Magma )  ->  ran  H  =  dom  dom  H
)
1914, 18sylan 469 . . . . 5  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  ran  H  =  dom  dom  H
)
2019raleqdv 3010 . . . 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 457 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  U  e.  dom  dom  H )
2322adantr 463 . . . . 5  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  U  e.  dom  dom  H )
2423, 19eleqtrrd 2493 . . . 4  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma )  ->  U  e.  ran  H )
254exidu1 25742 . . . . . . 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 451 . . . . 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 469 . . . 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 6285 . . . . . . . 8  |-  ( u  =  U  ->  (
u H x )  =  ( U H x ) )
3029eqeq1d 2404 . . . . . . 7  |-  ( u  =  U  ->  (
( u H x )  =  x  <->  ( U H x )  =  x ) )
31 oveq2 6286 . . . . . . . 8  |-  ( u  =  U  ->  (
x H u )  =  ( x H U ) )
3231eqeq1d 2404 . . . . . . 7  |-  ( u  =  U  ->  (
( x H u )  =  x  <->  ( x H U )  =  x ) )
3330, 32anbi12d 709 . . . . . 6  |-  ( u  =  U  ->  (
( ( u H x )  =  x  /\  ( x H u )  =  x )  <->  ( ( U H x )  =  x  /\  ( x H U )  =  x ) ) )
3433ralbidv 2843 . . . . 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 6262 . . . 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 659 . . 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 2443 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   E!wreu 2756   _Vcvv 3059    i^i cin 3413    C_ wss 3414    X. cxp 4821   dom cdm 4823   ran crn 4824    |` cres 4825   ` cfv 5569   iota_crio 6239  (class class class)co 6278  GIdcgi 25603    ExId cexid 25730   Magmacmagm 25734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fo 5575  df-fv 5577  df-riota 6240  df-ov 6281  df-gid 25608  df-exid 25731  df-mgmOLD 25735
This theorem is referenced by:  isdrngo2  31643
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