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Theorem exidreslem 28695
Description: Lemma for exidres 28696 and exidresid 28697. (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 5036 . . . . . . 7  |-  dom  H  =  dom  ( G  |`  ( Y  X.  Y
) )
3 xpss12 4940 . . . . . . . . . . 11  |-  ( ( Y  C_  X  /\  Y  C_  X )  -> 
( Y  X.  Y
)  C_  ( X  X.  X ) )
43anidms 645 . . . . . . . . . 10  |-  ( Y 
C_  X  ->  ( Y  X.  Y )  C_  ( X  X.  X
) )
5 exidres.1 . . . . . . . . . . . . 13  |-  X  =  ran  G
65opidon2 23762 . . . . . . . . . . . 12  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G : ( X  X.  X )
-onto-> X )
7 fof 5615 . . . . . . . . . . . 12  |-  ( G : ( X  X.  X ) -onto-> X  ->  G : ( X  X.  X ) --> X )
8 fdm 5558 . . . . . . . . . . . 12  |-  ( G : ( X  X.  X ) --> X  ->  dom  G  =  ( X  X.  X ) )
96, 7, 83syl 20 . . . . . . . . . . 11  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  dom  G  =  ( X  X.  X
) )
109sseq2d 3379 . . . . . . . . . 10  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( ( Y  X.  Y )  C_  dom  G  <->  ( Y  X.  Y )  C_  ( X  X.  X ) ) )
114, 10syl5ibr 221 . . . . . . . . 9  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( Y  C_  X  ->  ( Y  X.  Y )  C_  dom  G ) )
1211imp 429 . . . . . . . 8  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  ( Y  X.  Y )  C_  dom  G )
13 ssdmres 5127 . . . . . . . 8  |-  ( ( Y  X.  Y ) 
C_  dom  G  <->  dom  ( G  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
1412, 13sylib 196 . . . . . . 7  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  dom  ( G  |`  ( Y  X.  Y ) )  =  ( Y  X.  Y ) )
152, 14syl5eq 2482 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  dom  H  =  ( Y  X.  Y ) )
1615dmeqd 5037 . . . . 5  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  dom  dom 
H  =  dom  ( Y  X.  Y ) )
17 dmxpid 5054 . . . . 5  |-  dom  ( Y  X.  Y )  =  Y
1816, 17syl6eq 2486 . . . 4  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  dom  dom 
H  =  Y )
1918eleq2d 2505 . . 3  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  ( U  e.  dom  dom  H  <->  U  e.  Y ) )
2019biimp3ar 1319 . 2  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  U  e.  dom  dom  H )
21 ssel2 3346 . . . . . . . . . 10  |-  ( ( Y  C_  X  /\  x  e.  Y )  ->  x  e.  X )
22 exidres.2 . . . . . . . . . . 11  |-  U  =  (GId `  G )
235, 22cmpidelt 23767 . . . . . . . . . 10  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  x  e.  X )  ->  (
( U G x )  =  x  /\  ( x G U )  =  x ) )
2421, 23sylan2 474 . . . . . . . . 9  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  ( Y  C_  X  /\  x  e.  Y ) )  -> 
( ( U G x )  =  x  /\  ( x G U )  =  x ) )
2524anassrs 648 . . . . . . . 8  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X )  /\  x  e.  Y )  ->  ( ( U G x )  =  x  /\  ( x G U )  =  x ) )
2625adantrl 715 . . . . . . 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 6099 . . . . . . . . . . 11  |-  ( U H x )  =  ( U ( G  |`  ( Y  X.  Y
) ) x )
28 ovres 6225 . . . . . . . . . . 11  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( U ( G  |`  ( Y  X.  Y
) ) x )  =  ( U G x ) )
2927, 28syl5eq 2482 . . . . . . . . . 10  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( U H x )  =  ( U G x ) )
3029eqeq1d 2446 . . . . . . . . 9  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( ( U H x )  =  x  <-> 
( U G x )  =  x ) )
311oveqi 6099 . . . . . . . . . . . 12  |-  ( x H U )  =  ( x ( G  |`  ( Y  X.  Y
) ) U )
32 ovres 6225 . . . . . . . . . . . 12  |-  ( ( x  e.  Y  /\  U  e.  Y )  ->  ( x ( G  |`  ( Y  X.  Y
) ) U )  =  ( x G U ) )
3331, 32syl5eq 2482 . . . . . . . . . . 11  |-  ( ( x  e.  Y  /\  U  e.  Y )  ->  ( x H U )  =  ( x G U ) )
3433ancoms 453 . . . . . . . . . 10  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( x H U )  =  ( x G U ) )
3534eqeq1d 2446 . . . . . . . . 9  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( ( x H U )  =  x  <-> 
( x G U )  =  x ) )
3630, 35anbi12d 710 . . . . . . . 8  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( ( ( U H x )  =  x  /\  ( x H U )  =  x )  <->  ( ( U G x )  =  x  /\  ( x G U )  =  x ) ) )
3736adantl 466 . . . . . . 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 232 . . . . . 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 648 . . . . 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 2794 . . . 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 1182 . . 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 1008 . . . . . . . 8  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  ( Y  X.  Y )  C_  dom  G )
4342, 13sylib 196 . . . . . . 7  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  dom  ( G  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
442, 43syl5eq 2482 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  dom  H  =  ( Y  X.  Y
) )
4544dmeqd 5037 . . . . 5  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  dom  dom  H  =  dom  ( Y  X.  Y ) )
4645, 17syl6eq 2486 . . . 4  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  dom  dom  H  =  Y )
4746raleqdv 2918 . . 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 232 . 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 532 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710    i^i cin 3322    C_ wss 3323    X. cxp 4833   dom cdm 4835   ran crn 4836    |` cres 4837   -->wf 5409   -onto->wfo 5411   ` cfv 5413  (class class class)co 6086  GIdcgi 23625    ExId cexid 23752   Magmacmagm 23756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419  df-fv 5421  df-riota 6047  df-ov 6089  df-gid 23630  df-exid 23753  df-mgm 23757
This theorem is referenced by:  exidres  28696  exidresid  28697
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