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

Theorem s1fv 11647
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 11639 . . 3  |-  ( A  e.  B  ->  <" A ">  =  { <. 0 ,  A >. } )
21fveq1d 5634 . 2  |-  ( A  e.  B  ->  ( <" A "> `  0 )  =  ( { <. 0 ,  A >. } `  0 ) )
3 0nn0 10129 . . 3  |-  0  e.  NN0
4 fvsng 5827 . . 3  |-  ( ( 0  e.  NN0  /\  A  e.  B )  ->  ( { <. 0 ,  A >. } `  0
)  =  A )
53, 4mpan 651 . 2  |-  ( A  e.  B  ->  ( { <. 0 ,  A >. } `  0 )  =  A )
62, 5eqtrd 2398 1  |-  ( A  e.  B  ->  ( <" A "> `  0 )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647    e. wcel 1715   {csn 3729   <.cop 3732   ` cfv 5358   0cc0 8884   NN0cn0 10114   <"cs1 11606
This theorem is referenced by:  eqs1  11648  cats1un  11677  revs1  11684  cats1fvn  11709  s2fv0  11736  efgsval2  15252  efgs1  15254  efgsp1  15256  efgsfo  15258  pgpfaclem1  15526
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-mulcl 8946  ax-i2m1 8952
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-n0 10115  df-s1 11612
  Copyright terms: Public domain W3C validator