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Theorem exidreslem 26567
Description: Lemma for exidres 26568 and exidresid 26569. (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
exidreslem  |-  ( ( 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 ) ) )
Distinct variable groups:    x, G    x, Y    x, X    x, U    x, H

Proof of Theorem exidreslem
StepHypRef Expression
1 exidres.3 . . . . . . . 8  |-  H  =  ( G  |`  ( Y  X.  Y ) )
21dmeqi 4880 . . . . . . 7  |-  dom  H  =  dom  ( G  |`  ( Y  X.  Y
) )
3 xpss12 4792 . . . . . . . . . . 11  |-  ( ( Y  C_  X  /\  Y  C_  X )  -> 
( Y  X.  Y
)  C_  ( X  X.  X ) )
43anidms 626 . . . . . . . . . 10  |-  ( Y 
C_  X  ->  ( Y  X.  Y )  C_  ( X  X.  X
) )
5 exidres.1 . . . . . . . . . . . . 13  |-  X  =  ran  G
65opidon2 20991 . . . . . . . . . . . 12  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G : ( X  X.  X )
-onto-> X )
7 fof 5451 . . . . . . . . . . . 12  |-  ( G : ( X  X.  X ) -onto-> X  ->  G : ( X  X.  X ) --> X )
8 fdm 5393 . . . . . . . . . . . 12  |-  ( G : ( X  X.  X ) --> X  ->  dom  G  =  ( X  X.  X ) )
96, 7, 83syl 18 . . . . . . . . . . 11  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  dom  G  =  ( X  X.  X
) )
109sseq2d 3206 . . . . . . . . . 10  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( ( Y  X.  Y )  C_  dom  G  <->  ( Y  X.  Y )  C_  ( X  X.  X ) ) )
114, 10syl5ibr 212 . . . . . . . . 9  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( Y  C_  X  ->  ( Y  X.  Y )  C_  dom  G ) )
1211imp 418 . . . . . . . 8  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  ( Y  X.  Y )  C_  dom  G )
13 ssdmres 4977 . . . . . . . 8  |-  ( ( Y  X.  Y ) 
C_  dom  G  <->  dom  ( G  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
1412, 13sylib 188 . . . . . . 7  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  dom  ( G  |`  ( Y  X.  Y ) )  =  ( Y  X.  Y ) )
152, 14syl5eq 2327 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  dom  H  =  ( Y  X.  Y ) )
1615dmeqd 4881 . . . . 5  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  dom  dom 
H  =  dom  ( Y  X.  Y ) )
17 dmxpid 4898 . . . . 5  |-  dom  ( Y  X.  Y )  =  Y
1816, 17syl6eq 2331 . . . 4  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  dom  dom 
H  =  Y )
1918eleq2d 2350 . . 3  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  ( U  e.  dom  dom  H  <->  U  e.  Y ) )
2019biimp3ar 1282 . 2  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  U  e.  dom  dom  H )
21 ssel2 3175 . . . . . . . . . 10  |-  ( ( Y  C_  X  /\  x  e.  Y )  ->  x  e.  X )
22 exidres.2 . . . . . . . . . . 11  |-  U  =  (GId `  G )
235, 22cmpidelt 20996 . . . . . . . . . 10  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  x  e.  X )  ->  (
( U G x )  =  x  /\  ( x G U )  =  x ) )
2421, 23sylan2 460 . . . . . . . . 9  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  ( Y  C_  X  /\  x  e.  Y ) )  -> 
( ( U G x )  =  x  /\  ( x G U )  =  x ) )
2524anassrs 629 . . . . . . . 8  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X )  /\  x  e.  Y )  ->  ( ( U G x )  =  x  /\  ( x G U )  =  x ) )
2625adantrl 696 . . . . . . 7  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X )  /\  ( U  e.  Y  /\  x  e.  Y
) )  ->  (
( U G x )  =  x  /\  ( x G U )  =  x ) )
271oveqi 5871 . . . . . . . . . . 11  |-  ( U H x )  =  ( U ( G  |`  ( Y  X.  Y
) ) x )
28 ovres 5987 . . . . . . . . . . 11  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( U ( G  |`  ( Y  X.  Y
) ) x )  =  ( U G x ) )
2927, 28syl5eq 2327 . . . . . . . . . 10  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( U H x )  =  ( U G x ) )
3029eqeq1d 2291 . . . . . . . . 9  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( ( U H x )  =  x  <-> 
( U G x )  =  x ) )
311oveqi 5871 . . . . . . . . . . . 12  |-  ( x H U )  =  ( x ( G  |`  ( Y  X.  Y
) ) U )
32 ovres 5987 . . . . . . . . . . . 12  |-  ( ( x  e.  Y  /\  U  e.  Y )  ->  ( x ( G  |`  ( Y  X.  Y
) ) U )  =  ( x G U ) )
3331, 32syl5eq 2327 . . . . . . . . . . 11  |-  ( ( x  e.  Y  /\  U  e.  Y )  ->  ( x H U )  =  ( x G U ) )
3433ancoms 439 . . . . . . . . . 10  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( x H U )  =  ( x G U ) )
3534eqeq1d 2291 . . . . . . . . 9  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( ( x H U )  =  x  <-> 
( x G U )  =  x ) )
3630, 35anbi12d 691 . . . . . . . 8  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( ( ( U H x )  =  x  /\  ( x H U )  =  x )  <->  ( ( U G x )  =  x  /\  ( x G U )  =  x ) ) )
3736adantl 452 . . . . . . 7  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X )  /\  ( U  e.  Y  /\  x  e.  Y
) )  ->  (
( ( U H x )  =  x  /\  ( x H U )  =  x )  <->  ( ( U G x )  =  x  /\  ( x G U )  =  x ) ) )
3826, 37mpbird 223 . . . . . 6  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X )  /\  ( U  e.  Y  /\  x  e.  Y
) )  ->  (
( U H x )  =  x  /\  ( x H U )  =  x ) )
3938anassrs 629 . . . . 5  |-  ( ( ( ( G  e.  ( Magma  i^i  ExId  )  /\  Y  C_  X )  /\  U  e.  Y
)  /\  x  e.  Y )  ->  (
( U H x )  =  x  /\  ( x H U )  =  x ) )
4039ralrimiva 2626 . . . 4  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X )  /\  U  e.  Y )  ->  A. x  e.  Y  ( ( U H x )  =  x  /\  ( x H U )  =  x ) )
41403impa 1146 . . 3  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  A. x  e.  Y  ( ( U H x )  =  x  /\  ( x H U )  =  x ) )
42123adant3 975 . . . . . . . 8  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  ( Y  X.  Y )  C_  dom  G )
4342, 13sylib 188 . . . . . . 7  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  dom  ( G  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
442, 43syl5eq 2327 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  dom  H  =  ( Y  X.  Y
) )
4544dmeqd 4881 . . . . 5  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  dom  dom  H  =  dom  ( Y  X.  Y ) )
4645, 17syl6eq 2331 . . . 4  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  dom  dom  H  =  Y )
4746raleqdv 2742 . . 3  |-  ( ( 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 )  <->  A. x  e.  Y  ( ( U H x )  =  x  /\  ( x H U )  =  x ) ) )
4841, 47mpbird 223 . 2  |-  ( ( 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 ) )
4920, 48jca 518 1  |-  ( ( 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 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152    X. cxp 4687   dom cdm 4689   ran crn 4690    |` cres 4691   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858  GIdcgi 20854    ExId cexid 20981   Magmacmagm 20985
This theorem is referenced by:  exidres  26568  exidresid  26569
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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