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Theorem s1fv 11765
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 11757 . . 3  |-  ( A  e.  B  ->  <" A ">  =  { <. 0 ,  A >. } )
21fveq1d 5733 . 2  |-  ( A  e.  B  ->  ( <" A "> `  0 )  =  ( { <. 0 ,  A >. } `  0 ) )
3 0nn0 10241 . . 3  |-  0  e.  NN0
4 fvsng 5930 . . 3  |-  ( ( 0  e.  NN0  /\  A  e.  B )  ->  ( { <. 0 ,  A >. } `  0
)  =  A )
53, 4mpan 653 . 2  |-  ( A  e.  B  ->  ( { <. 0 ,  A >. } `  0 )  =  A )
62, 5eqtrd 2470 1  |-  ( A  e.  B  ->  ( <" A "> `  0 )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   {csn 3816   <.cop 3819   ` cfv 5457   0cc0 8995   NN0cn0 10226   <"cs1 11724
This theorem is referenced by:  eqs1  11766  cats1un  11795  revs1  11802  cats1fvn  11827  s2fv0  11854  efgsval2  15370  efgs1  15372  efgsp1  15374  efgsfo  15376  pgpfaclem1  15644  lstccats1fst  28297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-mulcl 9057  ax-i2m1 9063
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-n0 10227  df-s1 11730
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