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Theorem ununr 25420
Description: The unit of a unital ring is unique. (Contributed by FL, 12-Jul-2009.) (Proof shortened by Mario Carneiro, 23-Dec-2013.)
Hypotheses
Ref Expression
ununr.1  |-  H  =  ( 2nd `  R
)
ununr.2  |-  X  =  ran  ( 1st `  R
)
Assertion
Ref Expression
ununr  |-  ( R  e.  RingOps  ->  E! x  e.  X  A. y  e.  X  ( ( x H y )  =  y  /\  ( y H x )  =  y ) )
Distinct variable groups:    x, H, y    x, X, y
Allowed substitution hints:    R( x, y)

Proof of Theorem ununr
StepHypRef Expression
1 ununr.1 . . . 4  |-  H  =  ( 2nd `  R
)
21rngomndo 21088 . . 3  |-  ( R  e.  RingOps  ->  H  e. MndOp )
3 mndomgmid 21009 . . 3  |-  ( H  e. MndOp  ->  H  e.  (
Magma  i^i  ExId  ) )
4 eqid 2283 . . . 4  |-  ran  H  =  ran  H
54exidu1 20993 . . 3  |-  ( H  e.  ( Magma  i^i  ExId  )  ->  E! x  e. 
ran  H A. y  e.  ran  H ( ( x H y )  =  y  /\  (
y H x )  =  y ) )
62, 3, 53syl 18 . 2  |-  ( R  e.  RingOps  ->  E! x  e. 
ran  H A. y  e.  ran  H ( ( x H y )  =  y  /\  (
y H x )  =  y ) )
7 ununr.2 . . . 4  |-  X  =  ran  ( 1st `  R
)
8 eqid 2283 . . . . 5  |-  ( 1st `  R )  =  ( 1st `  R )
91, 8rngorn1eq 21087 . . . 4  |-  ( R  e.  RingOps  ->  ran  ( 1st `  R )  =  ran  H )
107, 9syl5req 2328 . . 3  |-  ( R  e.  RingOps  ->  ran  H  =  X )
11 raleq 2736 . . . 4  |-  ( ran 
H  =  X  -> 
( A. y  e. 
ran  H ( ( x H y )  =  y  /\  (
y H x )  =  y )  <->  A. y  e.  X  ( (
x H y )  =  y  /\  (
y H x )  =  y ) ) )
1211reueqd 2746 . . 3  |-  ( ran 
H  =  X  -> 
( E! x  e. 
ran  H A. y  e.  ran  H ( ( x H y )  =  y  /\  (
y H x )  =  y )  <->  E! x  e.  X  A. y  e.  X  ( (
x H y )  =  y  /\  (
y H x )  =  y ) ) )
1310, 12syl 15 . 2  |-  ( R  e.  RingOps  ->  ( E! x  e.  ran  H A. y  e.  ran  H ( ( x H y )  =  y  /\  (
y H x )  =  y )  <->  E! x  e.  X  A. y  e.  X  ( (
x H y )  =  y  /\  (
y H x )  =  y ) ) )
146, 13mpbid 201 1  |-  ( R  e.  RingOps  ->  E! x  e.  X  A. y  e.  X  ( ( x H y )  =  y  /\  ( y H x )  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E!wreu 2545    i^i cin 3151   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121    ExId cexid 20981   Magmacmagm 20985  MndOpcmndo 21004   RingOpscrngo 21042
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-pow 4188  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-rmo 2551  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-1st 6122  df-2nd 6123  df-grpo 20858  df-ablo 20949  df-ass 20980  df-exid 20982  df-mgm 20986  df-sgr 20998  df-mndo 21005  df-rngo 21043
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