Theorem s1eq 13380

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 6191	. . . 4 |- ( A = B -> ( _I ` A ) = ( _I ` B ) )
2	1	opeq2d 4409	. . 3 |- ( A = B -> <. 0 , ( _I ` A ) >. = <. 0 , ( _I ` B ) >. )
3	2	sneqd 4189	. 2 |- ( A = B -> { <. 0 , ( _I ` A ) >. } = { <. 0 , ( _I ` B ) >. } )
4		df-s1 13302	. 2 |- <" A "> = { <. 0 , ( _I ` A ) >. }
5		df-s1 13302	. 2 |- <" B "> = { <. 0 , ( _I ` B ) >. }
6	3, 4, 5	3eqtr4g 2681	1 |- ( A = B -> <" A "> = <" B "> )

Syntax hints:   -> wi 4   = wceq 1483   {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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  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-iota 5851  df-fv 5896  df-s1 13302

This theorem is referenced by:  s1eqd  13381  wrdl1exs1  13393  wrdl1s1  13394  wrdind  13476  wrd2ind  13477  ccats1swrdeqrex  13478  reuccats1lem  13479  reuccats1  13480  revs1  13514  vrmdval  17394  frgpup3lem  18190  vdegp1ci  26434  mrsubcv  31407  mrsubrn  31410  elmrsubrn  31417  mrsubvrs  31419  mvhval  31431  ccats1pfxeqrex  41422  reuccatpfxs1lem  41433  reuccatpfxs1  41434