Theorem strss 15910

Description: Propagate component extraction to a structure T from a subset structure S. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 15-Jan-2014.)

Hypotheses

Ref	Expression
strss.t	|- T e. _V
strss.f	|- Fun T
strss.s	|- S C_ T
strss.e	|- E = Slot ( E ` ndx )
strss.n	|- <. ( E ` ndx ) , C >. e. S

Assertion

Ref	Expression
strss	|- ( E ` T ) = ( E ` S )

Proof of Theorem strss

Step	Hyp	Ref	Expression
1		strss.e	. . 3 |- E = Slot ( E ` ndx )
2		strss.t	. . . 4 |- T e. _V
3	2	a1i 11	. . 3 |- ( T. -> T e. _V )
4		strss.f	. . . 4 |- Fun T
5	4	a1i 11	. . 3 |- ( T. -> Fun T )
6		strss.s	. . . 4 |- S C_ T
7	6	a1i 11	. . 3 |- ( T. -> S C_ T )
8		strss.n	. . . 4 |- <. ( E ` ndx ) , C >. e. S
9	8	a1i 11	. . 3 |- ( T. -> <. ( E ` ndx ) , C >. e. S )
10	1, 3, 5, 7, 9	strssd 15909	. 2 |- ( T. -> ( E ` T ) = ( E ` S ) )
11	10	trud 1493	1 |- ( E ` T ) = ( E ` S )

Syntax hints:   = wceq 1483   T. wtru 1484   e. wcel 1990   _Vcvv 3200   C_ wss 3574   <.cop 4183   Fun wfun 5882   ` cfv 5888   ndxcnx 15854  Slot cslot 15856

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-slot 15861

This theorem is referenced by:  grpss  17440