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Theorem exidres 25980
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 25979 . . 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 5865 . . . . . . 7  |-  ( u  =  U  ->  (
u H x )  =  ( U H x ) )
65eqeq1d 2291 . . . . . 6  |-  ( u  =  U  ->  (
( u H x )  =  x  <->  ( U H x )  =  x ) )
7 oveq2 5866 . . . . . . 7  |-  ( u  =  U  ->  (
x H u )  =  ( x H U ) )
87eqeq1d 2291 . . . . . 6  |-  ( u  =  U  ->  (
( x H u )  =  x  <->  ( x H U )  =  x ) )
96, 8anbi12d 691 . . . . 5  |-  ( u  =  U  ->  (
( ( u H x )  =  x  /\  ( x H u )  =  x )  <->  ( ( U H x )  =  x  /\  ( x H U )  =  x ) ) )
109ralbidv 2563 . . . 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 2884 . . 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 15 . 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 4994 . . . . 5  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( G  |`  ( Y  X.  Y
) )  e.  _V )
143, 13syl5eqel 2367 . . . 4  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  H  e.  _V )
15 eqid 2283 . . . . 5  |-  dom  dom  H  =  dom  dom  H
1615isexid 20984 . . . 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 15 . . 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 976 . 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 223 1  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  H  e.  ExId 
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    i^i cin 3151    C_ wss 3152    X. cxp 4687   dom cdm 4689   ran crn 4690    |` cres 4691   ` cfv 5255  (class class class)co 5858  GIdcgi 20854    ExId cexid 20981   Magmacmagm 20985
This theorem is referenced by:  exidresid  25981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-riota 6304  df-gid 20859  df-exid 20982  df-mgm 20986
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