Theorem s1val 13378

Description: Value of a single-symbol word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Assertion

Ref	Expression
s1val	|- ( A e. V -> <" A "> = { <. 0 , A >. } )

Proof of Theorem s1val

Step	Hyp	Ref	Expression
1		df-s1 13302	. 2 |- <" A "> = { <. 0 , ( _I ` A ) >. }
2		fvi 6255	. . . 4 |- ( A e. V -> ( _I ` A ) = A )
3	2	opeq2d 4409	. . 3 |- ( A e. V -> <. 0 , ( _I ` A ) >. = <. 0 , A >. )
4	3	sneqd 4189	. 2 |- ( A e. V -> { <. 0 , ( _I ` A ) >. } = { <. 0 , A >. } )
5	1, 4	syl5eq 2668	1 |- ( A e. V -> <" A "> = { <. 0 , A >. } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {csn 4177   <.cop 4183   _I cid 5023   ` cfv 5888   0cc0 9936   <"cs1 13294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-s1 13302

This theorem is referenced by:  s1rn  13379  s1cl  13382  s1dmALT  13389  s1fv  13390  s111  13395  repsw1  13530  s1co  13579  s2prop  13652  ofs1  13709  gsumws1  17376  uspgr1ewop  26140  usgr2v1e2w  26144  0wlkons1  26982  ofcs1  30621  signstf0  30645