Theorem s1eq 12792

Description: Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)

Assertion

Ref	Expression
s1eq	|- ( A = B -> <" A "> = <" B "> )

Proof of Theorem s1eq

Step	Hyp	Ref	Expression
1		fveq2 5879	. . . 4 |- ( A = B -> ( _I ` A ) = ( _I ` B ) )
2	1	opeq2d 4165	. . 3 |- ( A = B -> <. 0 , ( _I ` A ) >. = <. 0 , ( _I ` B ) >. )
3	2	sneqd 3971	. 2 |- ( A = B -> { <. 0 , ( _I ` A ) >. } = { <. 0 , ( _I ` B ) >. } )
4		df-s1 12714	. 2 |- <" A "> = { <. 0 , ( _I ` A ) >. }
5		df-s1 12714	. 2 |- <" B "> = { <. 0 , ( _I ` B ) >. }
6	3, 4, 5	3eqtr4g 2530	1 |- ( A = B -> <" A "> = <" B "> )

Syntax hints:   -> wi 4   = wceq 1452   {csn 3959   <.cop 3965   _I cid 4749   ` cfv 5589   0cc0 9557   <"cs1 12706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-iota 5553  df-fv 5597  df-s1 12714

This theorem is referenced by:  s1eqd  12793  wrdl1exs1  12804  wrdl1s1  12805  wrdind  12887  wrd2ind  12888  ccats1swrdeqrex  12889  reuccats1lem  12890  reuccats1  12891  revs1  12924  vrmdval  16719  frgpup3lem  17505  wwlkn0  25496  vdegp1ci  25793  mrsubcv  30220  mrsubrn  30223  elmrsubrn  30230  mrsubvrs  30232  mvhval  30244  ccats1pfxeqrex  39110  reuccatpfxs1lem  39121  reuccatpfxs1  39122  vdegp1ci-av  39761