Theorem strslss 13378

Description: Propagate component extraction to a structure T from a subset structure S. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Jim Kingdon, 31-Jan-2023.)

Hypotheses

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

Assertion

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

Proof of Theorem strslss

Step	Hyp	Ref	Expression
1		strslss.e	. . 3 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
2		strss.t	. . . 4 |- T e. _V
3	2	a1i 9	. . 3 |- ( T. -> T e. _V )
4		strss.f	. . . 4 |- Fun T
5	4	a1i 9	. . 3 |- ( T. -> Fun T )
6		strss.s	. . . 4 |- S C_ T
7	6	a1i 9	. . 3 |- ( T. -> S C_ T )
8		strss.n	. . . 4 |- <. ( E ` ndx ) , C >. e. S
9	8	a1i 9	. . 3 |- ( T. -> <. ( E ` ndx ) , C >. e. S )
10	1, 3, 5, 7, 9	strslssd 13377	. 2 |- ( T. -> ( E ` T ) = ( E ` S ) )
11	10	mptru 1411	1 |- ( E ` T ) = ( E ` S )

Syntax hints:   /\ wa 104   = wceq 1402   T. wtru 1403   e. wcel 2209   _Vcvv 2821   C_ wss 3220   <.cop 3708   Fun wfun 5366   ` cfv 5372   NNcn 9283   ndxcnx 13327  Slot cslot 13329

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 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-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fv 5380  df-slot 13334

This theorem is referenced by: (None)