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Theorem grpoidval 23856
Description: Lemma for grpoidcl 23857 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 23853 . . 3  |-  ( G  e.  GrpOp  ->  (GId `  G
)  =  ( iota_ u  e.  X  A. x  e.  X  ( (
u G x )  =  x  /\  (
x G u )  =  x ) ) )
4 simpl 457 . . . . . . . . 9  |-  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  ->  ( u G x )  =  x )
54ralimi 2819 . . . . . . . 8  |-  ( A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  ->  A. x  e.  X  ( u G x )  =  x )
65rgenw 2901 . . . . . . 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 11 . . . . . 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 23848 . . . . . . 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 457 . . . . . . . . 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 2819 . . . . . . . 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 2929 . . . . . . 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 16 . . . . . 6  |-  ( G  e.  GrpOp  ->  E. u  e.  X  A. x  e.  X  ( (
u G x )  =  x  /\  (
x G u )  =  x ) )
132grpoideu 23849 . . . . . 6  |-  ( G  e.  GrpOp  ->  E! u  e.  X  A. x  e.  X  ( u G x )  =  x )
147, 12, 133jca 1168 . . . . 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 3745 . . . . 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 471 . . . 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 6179 . . 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 2498 . 2  |-  ( G  e.  GrpOp  ->  (GId `  G
)  =  ( iota_ u  e.  X  A. x  e.  X  ( u G x )  =  x ) )
191, 18syl5eq 2507 1  |-  ( G  e.  GrpOp  ->  U  =  ( iota_ u  e.  X  A. x  e.  X  ( u G x )  =  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   E.wrex 2800   E!wreu 2801   ran crn 4950   ` cfv 5527   iota_crio 6161  (class class class)co 6201   GrpOpcgr 23826  GIdcgi 23827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fo 5533  df-fv 5535  df-riota 6162  df-ov 6204  df-grpo 23831  df-gid 23832
This theorem is referenced by:  grpoidcl  23857  grpoidinv2  23858  cnid  23991  mulid  23996  hilid  24716
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