MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  s1val Structured version   Unicode version

Theorem s1val 12662
Description: Value of a single-symbol word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Assertion
Ref Expression
s1val  |-  ( A  e.  V  ->  <" A ">  =  { <. 0 ,  A >. } )

Proof of Theorem s1val
StepHypRef Expression
1 df-s1 12592 . 2  |-  <" A ">  =  { <. 0 ,  (  _I  `  A ) >. }
2 fvi 5905 . . . 4  |-  ( A  e.  V  ->  (  _I  `  A )  =  A )
32opeq2d 4165 . . 3  |-  ( A  e.  V  ->  <. 0 ,  (  _I  `  A
) >.  =  <. 0 ,  A >. )
43sneqd 3983 . 2  |-  ( A  e.  V  ->  { <. 0 ,  (  _I  `  A ) >. }  =  { <. 0 ,  A >. } )
51, 4syl5eq 2455 1  |-  ( A  e.  V  ->  <" A ">  =  { <. 0 ,  A >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   {csn 3971   <.cop 3977    _I cid 4732   ` cfv 5568   0cc0 9521   <"cs1 12584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-s1 12592
This theorem is referenced by:  s1rn  12663  s1cl  12666  s1fv  12671  s111  12675  repsw1  12809  s1co  12853  s2prop  12916  gsumws1  16329  ofs1  28991  ofcs1  28992  signstf0  29017
  Copyright terms: Public domain W3C validator