Theorem s1eq 11195

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 5639	. . . 4 |- ( A = B -> ( _I ` A ) = ( _I ` B ) )
2	1	opeq2d 3869	. . 3 |- ( A = B -> <. 0 , ( _I ` A ) >. = <. 0 , ( _I ` B ) >. )
3	2	sneqd 3682	. 2 |- ( A = B -> { <. 0 , ( _I ` A ) >. } = { <. 0 , ( _I ` B ) >. } )
4		df-s1 11192	. 2 |- <" A "> = { <. 0 , ( _I ` A ) >. }
5		df-s1 11192	. 2 |- <" B "> = { <. 0 , ( _I ` B ) >. }
6	3, 4, 5	3eqtr4g 2289	1 |- ( A = B -> <" A "> = <" B "> )

Syntax hints:   -> wi 4   = wceq 1397   {csn 3669   <.cop 3672   _I cid 4385   ` cfv 5326   0cc0 8031   <"cs1 11191

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-s1 11192

This theorem is referenced by:  s1eqd  11196  wrdl1exs1  11205  wrdl1s1  11206  ccats1pfxeqrex  11295  wrdind  11302  wrd2ind  11303  reuccatpfxs1lem  11326  reuccatpfxs1  11327  vdegp1cid  16166