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Theorem s1fv 12409
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 12401 . . 3  |-  ( A  e.  B  ->  <" A ">  =  { <. 0 ,  A >. } )
21fveq1d 5794 . 2  |-  ( A  e.  B  ->  ( <" A "> `  0 )  =  ( { <. 0 ,  A >. } `  0 ) )
3 0nn0 10698 . . 3  |-  0  e.  NN0
4 fvsng 6014 . . 3  |-  ( ( 0  e.  NN0  /\  A  e.  B )  ->  ( { <. 0 ,  A >. } `  0
)  =  A )
53, 4mpan 670 . 2  |-  ( A  e.  B  ->  ( { <. 0 ,  A >. } `  0 )  =  A )
62, 5eqtrd 2492 1  |-  ( A  e.  B  ->  ( <" A "> `  0 )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   {csn 3978   <.cop 3984   ` cfv 5519   0cc0 9386   NN0cn0 10683   <"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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-mulcl 9448  ax-i2m1 9454
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-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  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-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-iota 5482  df-fun 5521  df-fv 5527  df-n0 10684  df-s1 12343
This theorem is referenced by:  lsws1  12410  eqs1  12411  wrdl1s1  12412  ccats1val2  12418  cats1un  12481  revs1  12516  cats1fvn  12596  s2fv0  12623  efgsval2  16343  efgs1  16345  efgsp1  16347  efgsfo  16349  pgpfaclem1  16696  signstf0  27106  signstfvn  27107  signsvtn0  27108  signstfvneq0  27110  ccat2s1p1  30406  ccat2s1p2  30407
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