MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  s1eq Structured version   Unicode version

Theorem s1eq 11753
Description: Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Assertion
Ref Expression
s1eq  |-  ( A  =  B  ->  <" A ">  =  <" B "> )

Proof of Theorem s1eq
StepHypRef Expression
1 fveq2 5728 . . . 4  |-  ( A  =  B  ->  (  _I  `  A )  =  (  _I  `  B
) )
21opeq2d 3991 . . 3  |-  ( A  =  B  ->  <. 0 ,  (  _I  `  A
) >.  =  <. 0 ,  (  _I  `  B
) >. )
32sneqd 3827 . 2  |-  ( A  =  B  ->  { <. 0 ,  (  _I  `  A ) >. }  =  { <. 0 ,  (  _I  `  B )
>. } )
4 df-s1 11725 . 2  |-  <" A ">  =  { <. 0 ,  (  _I  `  A ) >. }
5 df-s1 11725 . 2  |-  <" B ">  =  { <. 0 ,  (  _I  `  B ) >. }
63, 4, 53eqtr4g 2493 1  |-  ( A  =  B  ->  <" A ">  =  <" B "> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   {csn 3814   <.cop 3817    _I cid 4493   ` cfv 5454   0cc0 8990   <"cs1 11719
This theorem is referenced by:  s1eqd  11754  wrdind  11791  revs1  11797  vrmdval  14802  frgpup3lem  15409  vdegp1ci  21708
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
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-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  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-br 4213  df-iota 5418  df-fv 5462  df-s1 11725
  Copyright terms: Public domain W3C validator