Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  exidreslem Unicode version

Theorem exidreslem 26442
Description: Lemma for exidres 26443 and exidresid 26444. (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 5030 . . . . . . 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 627 . . . . . . . . . 10  |-  ( Y 
C_  X  ->  ( Y  X.  Y )  C_  ( X  X.  X
) )
5 exidres.1 . . . . . . . . . . . . 13  |-  X  =  ran  G
65opidon2 21865 . . . . . . . . . . . 12  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G : ( X  X.  X )
-onto-> X )
7 fof 5612 . . . . . . . . . . . 12  |-  ( G : ( X  X.  X ) -onto-> X  ->  G : ( X  X.  X ) --> X )
8 fdm 5554 . . . . . . . . . . . 12  |-  ( G : ( X  X.  X ) --> X  ->  dom  G  =  ( X  X.  X ) )
96, 7, 83syl 19 . . . . . . . . . . 11  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  dom  G  =  ( X  X.  X
) )
109sseq2d 3336 . . . . . . . . . 10  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( ( Y  X.  Y )  C_  dom  G  <->  ( Y  X.  Y )  C_  ( X  X.  X ) ) )
114, 10syl5ibr 213 . . . . . . . . 9  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ( Y  C_  X  ->  ( Y  X.  Y )  C_  dom  G ) )
1211imp 419 . . . . . . . 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 189 . . . . . . 7  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  dom  ( G  |`  ( Y  X.  Y ) )  =  ( Y  X.  Y ) )
152, 14syl5eq 2448 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  dom  H  =  ( Y  X.  Y ) )
1615dmeqd 5031 . . . . 5  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  dom  dom 
H  =  dom  ( Y  X.  Y ) )
17 dmxpid 5048 . . . . 5  |-  dom  ( Y  X.  Y )  =  Y
1816, 17syl6eq 2452 . . . 4  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  dom  dom 
H  =  Y )
1918eleq2d 2471 . . 3  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  ( U  e.  dom  dom  H  <->  U  e.  Y ) )
2019biimp3ar 1284 . 2  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  U  e.  dom  dom  H )
21 ssel2 3303 . . . . . . . . . 10  |-  ( ( Y  C_  X  /\  x  e.  Y )  ->  x  e.  X )
22 exidres.2 . . . . . . . . . . 11  |-  U  =  (GId `  G )
235, 22cmpidelt 21870 . . . . . . . . . 10  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  x  e.  X )  ->  (
( U G x )  =  x  /\  ( x G U )  =  x ) )
2421, 23sylan2 461 . . . . . . . . 9  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  ( Y  C_  X  /\  x  e.  Y ) )  -> 
( ( U G x )  =  x  /\  ( x G U )  =  x ) )
2524anassrs 630 . . . . . . . 8  |-  ( ( ( G  e.  (
Magma  i^i  ExId  )  /\  Y  C_  X )  /\  x  e.  Y )  ->  ( ( U G x )  =  x  /\  ( x G U )  =  x ) )
2625adantrl 697 . . . . . . 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 6053 . . . . . . . . . . 11  |-  ( U H x )  =  ( U ( G  |`  ( Y  X.  Y
) ) x )
28 ovres 6172 . . . . . . . . . . 11  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( U ( G  |`  ( Y  X.  Y
) ) x )  =  ( U G x ) )
2927, 28syl5eq 2448 . . . . . . . . . 10  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( U H x )  =  ( U G x ) )
3029eqeq1d 2412 . . . . . . . . 9  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( ( U H x )  =  x  <-> 
( U G x )  =  x ) )
311oveqi 6053 . . . . . . . . . . . 12  |-  ( x H U )  =  ( x ( G  |`  ( Y  X.  Y
) ) U )
32 ovres 6172 . . . . . . . . . . . 12  |-  ( ( x  e.  Y  /\  U  e.  Y )  ->  ( x ( G  |`  ( Y  X.  Y
) ) U )  =  ( x G U ) )
3331, 32syl5eq 2448 . . . . . . . . . . 11  |-  ( ( x  e.  Y  /\  U  e.  Y )  ->  ( x H U )  =  ( x G U ) )
3433ancoms 440 . . . . . . . . . 10  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( x H U )  =  ( x G U ) )
3534eqeq1d 2412 . . . . . . . . 9  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( ( x H U )  =  x  <-> 
( x G U )  =  x ) )
3630, 35anbi12d 692 . . . . . . . 8  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( ( ( U H x )  =  x  /\  ( x H U )  =  x )  <->  ( ( U G x )  =  x  /\  ( x G U )  =  x ) ) )
3736adantl 453 . . . . . . 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 224 . . . . . 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 630 . . . . 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 2749 . . . 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 1148 . . 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 977 . . . . . . . 8  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  ( Y  X.  Y )  C_  dom  G )
4342, 13sylib 189 . . . . . . 7  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  dom  ( G  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
442, 43syl5eq 2448 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  dom  H  =  ( Y  X.  Y
) )
4544dmeqd 5031 . . . . 5  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  dom  dom  H  =  dom  ( Y  X.  Y ) )
4645, 17syl6eq 2452 . . . 4  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  dom  dom  H  =  Y )
4746raleqdv 2870 . . 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 224 . 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 519 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666    i^i cin 3279    C_ wss 3280    X. cxp 4835   dom cdm 4837   ran crn 4838    |` cres 4839   -->wf 5409   -onto->wfo 5411   ` cfv 5413  (class class class)co 6040  GIdcgi 21728    ExId cexid 21855   Magmacmagm 21859
This theorem is referenced by:  exidres  26443  exidresid  26444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419  df-fv 5421  df-ov 6043  df-riota 6508  df-gid 21733  df-exid 21856  df-mgm 21860
  Copyright terms: Public domain W3C validator