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Theorem s1fv 12578
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 12570 . . 3  |-  ( A  e.  B  ->  <" A ">  =  { <. 0 ,  A >. } )
21fveq1d 5866 . 2  |-  ( A  e.  B  ->  ( <" A "> `  0 )  =  ( { <. 0 ,  A >. } `  0 ) )
3 0nn0 10806 . . 3  |-  0  e.  NN0
4 fvsng 6093 . . 3  |-  ( ( 0  e.  NN0  /\  A  e.  B )  ->  ( { <. 0 ,  A >. } `  0
)  =  A )
53, 4mpan 670 . 2  |-  ( A  e.  B  ->  ( { <. 0 ,  A >. } `  0 )  =  A )
62, 5eqtrd 2508 1  |-  ( A  e.  B  ->  ( <" A "> `  0 )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {csn 4027   <.cop 4033   ` cfv 5586   0cc0 9488   NN0cn0 10791   <"cs1 12499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-mulcl 9550  ax-i2m1 9556
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-n0 10792  df-s1 12507
This theorem is referenced by:  lsws1  12579  eqs1  12580  wrdl1s1  12581  ccats1val2  12590  ccat2s1p1  12591  ccat2s1p2  12592  cats1un  12660  revs1  12698  cats1fvn  12782  s2fv0  12809  efgsval2  16547  efgs1  16549  efgsp1  16551  efgsfo  16553  pgpfaclem1  16922  signstf0  28165  signstfvn  28166  signsvtn0  28167  signstfvneq0  28169
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