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Theorem grpoidval 20899
Description: Lemma for grpoidcl 20900 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 20896 . . 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 2631 . . . . . . . 8  |-  ( A. x  e.  X  (
( u G x )  =  x  /\  ( x G u )  =  x )  ->  A. x  e.  X  ( u G x )  =  x )
65rgenw 2623 . . . . . . 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 20891 . . . . . . 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 2631 . . . . . . . 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 2663 . . . . . . 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 20892 . . . . . 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 3467 . . . . 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 6337 . . 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 2331 . 2  |-  ( G  e.  GrpOp  ->  (GId `  G
)  =  ( iota_ u  e.  X A. x  e.  X  ( u G x )  =  x ) )
191, 18syl5eq 2340 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   E!wreu 2558   ran crn 4706   ` cfv 5271  (class class class)co 5874   iota_crio 6313   GrpOpcgr 20869  GIdcgi 20870
This theorem is referenced by:  grpoidcl  20900  grpoidinv2  20901  cnid  21034  mulid  21039  hilid  21756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-riota 6320  df-grpo 20874  df-gid 20875
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