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Theorem s1eq 12402
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 5792 . . . 4  |-  ( A  =  B  ->  (  _I  `  A )  =  (  _I  `  B
) )
21opeq2d 4167 . . 3  |-  ( A  =  B  ->  <. 0 ,  (  _I  `  A
) >.  =  <. 0 ,  (  _I  `  B
) >. )
32sneqd 3990 . 2  |-  ( A  =  B  ->  { <. 0 ,  (  _I  `  A ) >. }  =  { <. 0 ,  (  _I  `  B )
>. } )
4 df-s1 12343 . 2  |-  <" A ">  =  { <. 0 ,  (  _I  `  A ) >. }
5 df-s1 12343 . 2  |-  <" B ">  =  { <. 0 ,  (  _I  `  B ) >. }
63, 4, 53eqtr4g 2517 1  |-  ( A  =  B  ->  <" A ">  =  <" B "> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370   {csn 3978   <.cop 3984    _I cid 4732   ` cfv 5519   0cc0 9386   <"cs1 12335
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-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
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-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-iota 5482  df-fv 5527  df-s1 12343
This theorem is referenced by:  s1eqd  12403  wrdl1s1  12412  wrdind  12482  wrd2ind  12483  revs1  12516  vrmdval  15646  frgpup3lem  16387  vdegp1ci  23752  signstfveq0  27115  ccats1swrdeqrex  30400  ccats1rev  30401  reuccats1  30402  wwlkn0  30464
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