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Theorem exidres 29930
Description: The restriction of a binary operation with identity to a subset containing the identity has an 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
exidres  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  H  e.  ExId 
)

Proof of Theorem exidres
Dummy variables  x  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exidres.1 . . . 4  |-  X  =  ran  G
2 exidres.2 . . . 4  |-  U  =  (GId `  G )
3 exidres.3 . . . 4  |-  H  =  ( G  |`  ( Y  X.  Y ) )
41, 2, 3exidreslem 29929 . . 3  |-  ( ( 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 ) ) )
5 oveq1 6282 . . . . . . 7  |-  ( u  =  U  ->  (
u H x )  =  ( U H x ) )
65eqeq1d 2462 . . . . . 6  |-  ( u  =  U  ->  (
( u H x )  =  x  <->  ( U H x )  =  x ) )
7 oveq2 6283 . . . . . . 7  |-  ( u  =  U  ->  (
x H u )  =  ( x H U ) )
87eqeq1d 2462 . . . . . 6  |-  ( u  =  U  ->  (
( x H u )  =  x  <->  ( x H U )  =  x ) )
96, 8anbi12d 710 . . . . 5  |-  ( u  =  U  ->  (
( ( u H x )  =  x  /\  ( x H u )  =  x )  <->  ( ( U H x )  =  x  /\  ( x H U )  =  x ) ) )
109ralbidv 2896 . . . 4  |-  ( u  =  U  ->  ( A. x  e.  dom  dom 
H ( ( u H x )  =  x  /\  ( x H u )  =  x )  <->  A. x  e.  dom  dom  H (
( U H x )  =  x  /\  ( x H U )  =  x ) ) )
1110rspcev 3207 . . 3  |-  ( ( U  e.  dom  dom  H  /\  A. x  e. 
dom  dom  H ( ( U H x )  =  x  /\  (
x H U )  =  x ) )  ->  E. u  e.  dom  dom 
H A. x  e. 
dom  dom  H ( ( u H x )  =  x  /\  (
x H u )  =  x ) )
124, 11syl 16 . 2  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  E. u  e.  dom  dom  H A. x  e.  dom  dom  H
( ( u H x )  =  x  /\  ( x H u )  =  x ) )
13 resexg 5307 . . . . 5  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( G  |`  ( Y  X.  Y
) )  e.  _V )
143, 13syl5eqel 2552 . . . 4  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  H  e.  _V )
15 eqid 2460 . . . . 5  |-  dom  dom  H  =  dom  dom  H
1615isexid 24981 . . . 4  |-  ( H  e.  _V  ->  ( H  e.  ExId  <->  E. u  e.  dom  dom  H A. x  e.  dom  dom  H
( ( u H x )  =  x  /\  ( x H u )  =  x ) ) )
1714, 16syl 16 . . 3  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( H  e. 
ExId 
<->  E. u  e.  dom  dom 
H A. x  e. 
dom  dom  H ( ( u H x )  =  x  /\  (
x H u )  =  x ) ) )
18173ad2ant1 1012 . 2  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  ( H  e.  ExId  <->  E. u  e.  dom  dom 
H A. x  e. 
dom  dom  H ( ( u H x )  =  x  /\  (
x H u )  =  x ) ) )
1912, 18mpbird 232 1  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  H  e.  ExId 
)
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.wrex 2808   _Vcvv 3106    i^i cin 3468    C_ wss 3469    X. cxp 4990   dom cdm 4992   ran crn 4993    |` cres 4994   ` cfv 5579  (class class class)co 6275  GIdcgi 24851    ExId cexid 24978   Magmacmagm 24982
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 24856  df-exid 24979  df-mgm 24983
This theorem is referenced by:  exidresid  29931
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