Theorem s1eq 12402

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 5792	. . . 4 |- ( A = B -> ( _I ` A ) = ( _I ` B ) )
2	1	opeq2d 4167	. . 3 |- ( A = B -> <. 0 , ( _I ` A ) >. = <. 0 , ( _I ` B ) >. )
3	2	sneqd 3990	. 2 |- ( A = B -> { <. 0 , ( _I ` A ) >. } = { <. 0 , ( _I ` B ) >. } )
4		df-s1 12343	. 2 |- <" A "> = { <. 0 , ( _I ` A ) >. }
5		df-s1 12343	. 2 |- <" B "> = { <. 0 , ( _I ` B ) >. }
6	3, 4, 5	3eqtr4g 2517	1 |- ( A = B -> <" A "> = <" B "> )

Syntax hints:   -> wi 4   = wceq 1370   {csn 3978   <.cop 3984   _I cid 4732   ` cfv 5519   0cc0 9386   <"cs1 12335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-iota 5482  df-fv 5527  df-s1 12343

This theorem is referenced by:  s1eqd  12403  wrdl1s1  12412  wrdind  12482  wrd2ind  12483  revs1  12516  vrmdval  15646  frgpup3lem  16387  vdegp1ci  23752  signstfveq0  27115  ccats1swrdeqrex  30400  ccats1rev  30401  reuccats1  30402  wwlkn0  30464