MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  s1fv Structured version   Unicode version

Theorem s1fv 12580
Description: Sole symbol of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Assertion
Ref Expression
s1fv  |-  ( A  e.  B  ->  ( <" A "> `  0 )  =  A )

Proof of Theorem s1fv
StepHypRef Expression
1 s1val 12571 . . 3  |-  ( A  e.  B  ->  <" A ">  =  { <. 0 ,  A >. } )
21fveq1d 5807 . 2  |-  ( A  e.  B  ->  ( <" A "> `  0 )  =  ( { <. 0 ,  A >. } `  0 ) )
3 0nn0 10771 . . 3  |-  0  e.  NN0
4 fvsng 6041 . . 3  |-  ( ( 0  e.  NN0  /\  A  e.  B )  ->  ( { <. 0 ,  A >. } `  0
)  =  A )
53, 4mpan 668 . 2  |-  ( A  e.  B  ->  ( { <. 0 ,  A >. } `  0 )  =  A )
62, 5eqtrd 2443 1  |-  ( A  e.  B  ->  ( <" A "> `  0 )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   {csn 3971   <.cop 3977   ` cfv 5525   0cc0 9442   NN0cn0 10756   <"cs1 12493
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  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-mulcl 9504  ax-i2m1 9510
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 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-iota 5489  df-fun 5527  df-fv 5533  df-n0 10757  df-s1 12501
This theorem is referenced by:  lsws1  12581  eqs1  12582  wrdl1s1  12583  ccats1val2  12592  ccat2s1p1  12593  ccat2s1p2  12594  cats1un  12664  revs1  12702  cats1fvn  12786  s2fv0  12813  efgsval2  16967  efgs1  16969  efgsp1  16971  efgsfo  16973  pgpfaclem1  17344  signstf0  28911  signstfvn  28912  signsvtn0  28913  signstfvneq0  28915
  Copyright terms: Public domain W3C validator