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Theorem exidres 26553
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 26552 . . 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 6088 . . . . . . 7  |-  ( u  =  U  ->  (
u H x )  =  ( U H x ) )
65eqeq1d 2444 . . . . . 6  |-  ( u  =  U  ->  (
( u H x )  =  x  <->  ( U H x )  =  x ) )
7 oveq2 6089 . . . . . . 7  |-  ( u  =  U  ->  (
x H u )  =  ( x H U ) )
87eqeq1d 2444 . . . . . 6  |-  ( u  =  U  ->  (
( x H u )  =  x  <->  ( x H U )  =  x ) )
96, 8anbi12d 692 . . . . 5  |-  ( u  =  U  ->  (
( ( u H x )  =  x  /\  ( x H u )  =  x )  <->  ( ( U H x )  =  x  /\  ( x H U )  =  x ) ) )
109ralbidv 2725 . . . 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 3052 . . 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 5185 . . . . 5  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( G  |`  ( Y  X.  Y
) )  e.  _V )
143, 13syl5eqel 2520 . . . 4  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  H  e.  _V )
15 eqid 2436 . . . . 5  |-  dom  dom  H  =  dom  dom  H
1615isexid 21905 . . . 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 978 . 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 224 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   _Vcvv 2956    i^i cin 3319    C_ wss 3320    X. cxp 4876   dom cdm 4878   ran crn 4879    |` cres 4880   ` cfv 5454  (class class class)co 6081  GIdcgi 21775    ExId cexid 21902   Magmacmagm 21906
This theorem is referenced by:  exidresid  26554
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-ov 6084  df-riota 6549  df-gid 21780  df-exid 21903  df-mgm 21907
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