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Theorem exidreslem 29942
Description: Lemma for exidres 29943 and exidresid 29944. (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 5202 . . . . . . 7  |-  dom  H  =  dom  ( G  |`  ( Y  X.  Y
) )
3 xpss12 5106 . . . . . . . . . . 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 25002 . . . . . . . . . . . 12  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G : ( X  X.  X )
-onto-> X )
7 fof 5793 . . . . . . . . . . . 12  |-  ( G : ( X  X.  X ) -onto-> X  ->  G : ( X  X.  X ) --> X )
8 fdm 5733 . . . . . . . . . . . 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 3532 . . . . . . . . . 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 5293 . . . . . . . 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 2520 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  dom  H  =  ( Y  X.  Y ) )
1615dmeqd 5203 . . . . 5  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  dom  dom 
H  =  dom  ( Y  X.  Y ) )
17 dmxpid 5220 . . . . 5  |-  dom  ( Y  X.  Y )  =  Y
1816, 17syl6eq 2524 . . . 4  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  dom  dom 
H  =  Y )
1918eleq2d 2537 . . 3  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  ( U  e.  dom  dom  H  <->  U  e.  Y ) )
2019biimp3ar 1329 . 2  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  U  e.  dom  dom  H )
21 ssel2 3499 . . . . . . . . . 10  |-  ( ( Y  C_  X  /\  x  e.  Y )  ->  x  e.  X )
22 exidres.2 . . . . . . . . . . 11  |-  U  =  (GId `  G )
235, 22cmpidelt 25007 . . . . . . . . . 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 6295 . . . . . . . . . . 11  |-  ( U H x )  =  ( U ( G  |`  ( Y  X.  Y
) ) x )
28 ovres 6424 . . . . . . . . . . 11  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( U ( G  |`  ( Y  X.  Y
) ) x )  =  ( U G x ) )
2927, 28syl5eq 2520 . . . . . . . . . 10  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( U H x )  =  ( U G x ) )
3029eqeq1d 2469 . . . . . . . . 9  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( ( U H x )  =  x  <-> 
( U G x )  =  x ) )
311oveqi 6295 . . . . . . . . . . . 12  |-  ( x H U )  =  ( x ( G  |`  ( Y  X.  Y
) ) U )
32 ovres 6424 . . . . . . . . . . . 12  |-  ( ( x  e.  Y  /\  U  e.  Y )  ->  ( x ( G  |`  ( Y  X.  Y
) ) U )  =  ( x G U ) )
3331, 32syl5eq 2520 . . . . . . . . . . 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 2469 . . . . . . . . 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 2878 . . . 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 1191 . . 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 1016 . . . . . . . 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 2520 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  dom  H  =  ( Y  X.  Y
) )
4544dmeqd 5203 . . . . 5  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  dom  dom  H  =  dom  ( Y  X.  Y ) )
4645, 17syl6eq 2524 . . . 4  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  dom  dom  H  =  Y )
4746raleqdv 3064 . . 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 973    = wceq 1379    e. wcel 1767   A.wral 2814    i^i cin 3475    C_ wss 3476    X. cxp 4997   dom cdm 4999   ran crn 5000    |` cres 5001   -->wf 5582   -onto->wfo 5584   ` cfv 5586  (class class class)co 6282  GIdcgi 24865    ExId cexid 24992   Magmacmagm 24996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fo 5592  df-fv 5594  df-riota 6243  df-ov 6285  df-gid 24870  df-exid 24993  df-mgm 24997
This theorem is referenced by:  exidres  29943  exidresid  29944
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