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Theorem grpoidval 20883
Description: Lemma for grpoidcl 20884 and others. (Contributed by NM, 5-Feb-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpoidval.1  |-  X  =  ran  G
grpoidval.2  |-  U  =  (GId `  G )
Assertion
Ref Expression
grpoidval  |-  ( G  e.  GrpOp  ->  U  =  ( iota_ u  e.  X A. x  e.  X  ( u G x )  =  x ) )
Distinct variable groups:    x, u, G    u, U, x    u, X, x

Proof of Theorem grpoidval
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 grpoidval.2 . 2  |-  U  =  (GId `  G )
2 grpoidval.1 . . . 4  |-  X  =  ran  G
32gidval 20880 . . 3  |-  ( G  e.  GrpOp  ->  (GId `  G
)  =  ( iota_ u  e.  X A. x  e.  X  ( (
u G x )  =  x  /\  (
x G u )  =  x ) ) )
4 simpl 443 . . . . . . . . 9  |-  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  ( u G x )  =  x )
54ralimi 2618 . . . . . . . 8  |-  ( A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  ->  A. x  e.  X  ( u G x )  =  x )
65rgenw 2610 . . . . . . 7  |-  A. u  e.  X  ( A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  ->  A. x  e.  X  ( u G x )  =  x )
76a1i 10 . . . . . 6  |-  ( G  e.  GrpOp  ->  A. u  e.  X  ( A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  ->  A. x  e.  X  ( u G x )  =  x ) )
82grpoidinv 20875 . . . . . . 7  |-  ( G  e.  GrpOp  ->  E. u  e.  X  A. x  e.  X  ( (
( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( ( y G x )  =  u  /\  ( x G y )  =  u ) ) )
9 simpl 443 . . . . . . . . 9  |-  ( ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( (
y G x )  =  u  /\  (
x G y )  =  u ) )  ->  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )
109ralimi 2618 . . . . . . . 8  |-  ( A. x  e.  X  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( (
y G x )  =  u  /\  (
x G y )  =  u ) )  ->  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )
1110reximi 2650 . . . . . . 7  |-  ( E. u  e.  X  A. x  e.  X  (
( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  ( (
y G x )  =  u  /\  (
x G y )  =  u ) )  ->  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )
128, 11syl 15 . . . . . 6  |-  ( G  e.  GrpOp  ->  E. u  e.  X  A. x  e.  X  ( (
u G x )  =  x  /\  (
x G u )  =  x ) )
132grpoideu 20876 . . . . . 6  |-  ( G  e.  GrpOp  ->  E! u  e.  X  A. x  e.  X  ( u G x )  =  x )
147, 12, 133jca 1132 . . . . 5  |-  ( G  e.  GrpOp  ->  ( A. u  e.  X  ( A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  A. x  e.  X  ( u G x )  =  x )  /\  E. u  e.  X  A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  /\  E! u  e.  X  A. x  e.  X  ( u G x )  =  x ) )
15 reupick2 3454 . . . . 5  |-  ( ( ( A. u  e.  X  ( A. x  e.  X  ( (
u G x )  =  x  /\  (
x G u )  =  x )  ->  A. x  e.  X  ( u G x )  =  x )  /\  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E! u  e.  X  A. x  e.  X  (
u G x )  =  x )  /\  u  e.  X )  ->  ( A. x  e.  X  ( u G x )  =  x  <->  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
1614, 15sylan 457 . . . 4  |-  ( ( G  e.  GrpOp  /\  u  e.  X )  ->  ( A. x  e.  X  ( u G x )  =  x  <->  A. x  e.  X  ( (
u G x )  =  x  /\  (
x G u )  =  x ) ) )
1716riotabidva 6321 . . 3  |-  ( G  e.  GrpOp  ->  ( iota_ u  e.  X A. x  e.  X  ( u G x )  =  x )  =  (
iota_ u  e.  X A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
183, 17eqtr4d 2318 . 2  |-  ( G  e.  GrpOp  ->  (GId `  G
)  =  ( iota_ u  e.  X A. x  e.  X  ( u G x )  =  x ) )
191, 18syl5eq 2327 1  |-  ( G  e.  GrpOp  ->  U  =  ( iota_ u  e.  X A. x  e.  X  ( u G x )  =  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   E.wrex 2544   E!wreu 2545   ran crn 4690   ` cfv 5255  (class class class)co 5858   iota_crio 6297   GrpOpcgr 20853  GIdcgi 20854
This theorem is referenced by:  grpoidcl  20884  grpoidinv2  20885  cnid  21018  mulid  21023  hilid  21740
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-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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-riota 6304  df-grpo 20858  df-gid 20859
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