Theorem s1eq 11245

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 5648	. . . 4 |- ( A = B -> ( _I ` A ) = ( _I ` B ) )
2	1	opeq2d 3874	. . 3 |- ( A = B -> <. 0 , ( _I ` A ) >. = <. 0 , ( _I ` B ) >. )
3	2	sneqd 3686	. 2 |- ( A = B -> { <. 0 , ( _I ` A ) >. } = { <. 0 , ( _I ` B ) >. } )
4		df-s1 11242	. 2 |- <" A "> = { <. 0 , ( _I ` A ) >. }
5		df-s1 11242	. 2 |- <" B "> = { <. 0 , ( _I ` B ) >. }
6	3, 4, 5	3eqtr4g 2289	1 |- ( A = B -> <" A "> = <" B "> )

Syntax hints:   -> wi 4   = wceq 1398   {csn 3673   <.cop 3676   _I cid 4391   ` cfv 5333   0cc0 8075   <"cs1 11241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-s1 11242

This theorem is referenced by:  s1eqd  11246  wrdl1exs1  11255  wrdl1s1  11256  ccats1pfxeqrex  11345  wrdind  11352  wrd2ind  11353  reuccatpfxs1lem  11376  reuccatpfxs1  11377  vdegp1cid  16240