Theorem s1eq 12601

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 5848	. . . 4 |- ( A = B -> ( _I ` A ) = ( _I ` B ) )
2	1	opeq2d 4210	. . 3 |- ( A = B -> <. 0 , ( _I ` A ) >. = <. 0 , ( _I ` B ) >. )
3	2	sneqd 4028	. 2 |- ( A = B -> { <. 0 , ( _I ` A ) >. } = { <. 0 , ( _I ` B ) >. } )
4		df-s1 12529	. 2 |- <" A "> = { <. 0 , ( _I ` A ) >. }
5		df-s1 12529	. 2 |- <" B "> = { <. 0 , ( _I ` B ) >. }
6	3, 4, 5	3eqtr4g 2520	1 |- ( A = B -> <" A "> = <" B "> )

Syntax hints:   -> wi 4   = wceq 1398   {csn 4016   <.cop 4022   _I cid 4779   ` cfv 5570   0cc0 9481   <"cs1 12521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-s1 12529

This theorem is referenced by:  s1eqd  12602  wrdl1s1  12611  wrdind  12693  wrd2ind  12694  ccats1swrdeqrex  12695  reuccats1lem  12696  reuccats1  12697  revs1  12730  vrmdval  16224  frgpup3lem  16994  wwlkn0  24891  vdegp1ci  25188  mrsubcv  29134  mrsubrn  29137  elmrsubrn  29144  mrsubvrs  29146  mvhval  29158  ccats1pfxeqrex  32650  reuccatpfxs1lem  32661  reuccatpfxs1  32662