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Theorem exidreslem 26552
Description: Lemma for exidres 26553 and exidresid 26554. (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 5071 . . . . . . 7  |-  dom  H  =  dom  ( G  |`  ( Y  X.  Y
) )
3 xpss12 4981 . . . . . . . . . . 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 21912 . . . . . . . . . . . 12  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G : ( X  X.  X )
-onto-> X )
7 fof 5653 . . . . . . . . . . . 12  |-  ( G : ( X  X.  X ) -onto-> X  ->  G : ( X  X.  X ) --> X )
8 fdm 5595 . . . . . . . . . . . 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 3376 . . . . . . . . . 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 5168 . . . . . . . 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 2480 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  dom  H  =  ( Y  X.  Y ) )
1615dmeqd 5072 . . . . 5  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  dom  dom 
H  =  dom  ( Y  X.  Y ) )
17 dmxpid 5089 . . . . 5  |-  dom  ( Y  X.  Y )  =  Y
1816, 17syl6eq 2484 . . . 4  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X )  ->  dom  dom 
H  =  Y )
1918eleq2d 2503 . . 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 3343 . . . . . . . . . 10  |-  ( ( Y  C_  X  /\  x  e.  Y )  ->  x  e.  X )
22 exidres.2 . . . . . . . . . . 11  |-  U  =  (GId `  G )
235, 22cmpidelt 21917 . . . . . . . . . 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 6094 . . . . . . . . . . 11  |-  ( U H x )  =  ( U ( G  |`  ( Y  X.  Y
) ) x )
28 ovres 6213 . . . . . . . . . . 11  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( U ( G  |`  ( Y  X.  Y
) ) x )  =  ( U G x ) )
2927, 28syl5eq 2480 . . . . . . . . . 10  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( U H x )  =  ( U G x ) )
3029eqeq1d 2444 . . . . . . . . 9  |-  ( ( U  e.  Y  /\  x  e.  Y )  ->  ( ( U H x )  =  x  <-> 
( U G x )  =  x ) )
311oveqi 6094 . . . . . . . . . . . 12  |-  ( x H U )  =  ( x ( G  |`  ( Y  X.  Y
) ) U )
32 ovres 6213 . . . . . . . . . . . 12  |-  ( ( x  e.  Y  /\  U  e.  Y )  ->  ( x ( G  |`  ( Y  X.  Y
) ) U )  =  ( x G U ) )
3331, 32syl5eq 2480 . . . . . . . . . . 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 2444 . . . . . . . . 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 2789 . . . 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 2480 . . . . . 6  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  dom  H  =  ( Y  X.  Y
) )
4544dmeqd 5072 . . . . 5  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  dom  dom  H  =  dom  ( Y  X.  Y ) )
4645, 17syl6eq 2484 . . . 4  |-  ( ( G  e.  ( Magma  i^i 
ExId  )  /\  Y  C_  X  /\  U  e.  Y
)  ->  dom  dom  H  =  Y )
4746raleqdv 2910 . . 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 1652    e. wcel 1725   A.wral 2705    i^i cin 3319    C_ wss 3320    X. cxp 4876   dom cdm 4878   ran crn 4879    |` cres 4880   -->wf 5450   -onto->wfo 5452   ` cfv 5454  (class class class)co 6081  GIdcgi 21775    ExId cexid 21902   Magmacmagm 21906
This theorem is referenced by:  exidres  26553  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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