Theorem s1eq 11387

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 5695	. . . 4 |- ( A = B -> ( _I ` A ) = ( _I ` B ) )
2	1	opeq2d 3911	. . 3 |- ( A = B -> <. 0 , ( _I ` A ) >. = <. 0 , ( _I ` B ) >. )
3	2	sneqd 3722	. 2 |- ( A = B -> { <. 0 , ( _I ` A ) >. } = { <. 0 , ( _I ` B ) >. } )
4		df-s1 11384	. 2 |- <" A "> = { <. 0 , ( _I ` A ) >. }
5		df-s1 11384	. 2 |- <" B "> = { <. 0 , ( _I ` B ) >. }
6	3, 4, 5	3eqtr4g 2296	1 |- ( A = B -> <" A "> = <" B "> )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   {csn 3709   <.cop 3712   _I cid 4433   ` cfv 5377   0cc0 8179   <"cs1 11383

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-s1 11384

This theorem is used by:  s1eqd  11388  wrdl1exs1  11397  wrdl1s1  11398  ccats1pfxeqrex  11487  wrdind  11494  wrd2ind  11495  reuccatpfxs1lem  11518  reuccatpfxs1  11519  vdegp1cid  16557