Theorem strslssd 12994

Description: Deduction version of strslss 12995. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 31-Jan-2023.)

Hypotheses

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

Assertion

Ref	Expression
strslssd	|- ( ph -> ( E ` T ) = ( E ` S ) )

Proof of Theorem strslssd

Step	Hyp	Ref	Expression
1		strslssd.e	. . 3 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
2		strssd.t	. . 3 |- ( ph -> T e. V )
3		strssd.f	. . 3 |- ( ph -> Fun T )
4		strssd.s	. . . 4 |- ( ph -> S C_ T )
5		strssd.n	. . . 4 |- ( ph -> <. ( E ` ndx ) , C >. e. S )
6	4, 5	sseldd 3202	. . 3 |- ( ph -> <. ( E ` ndx ) , C >. e. T )
7	1, 2, 3, 6	strslfvd 12989	. 2 |- ( ph -> C = ( E ` T ) )
8	2, 4	ssexd 4200	. . 3 |- ( ph -> S e. _V )
9		funss 5309	. . . 4 |- ( S C_ T -> ( Fun T -> Fun S ) )
10	4, 3, 9	sylc 62	. . 3 |- ( ph -> Fun S )
11	1, 8, 10, 5	strslfvd 12989	. 2 |- ( ph -> C = ( E ` S ) )
12	7, 11	eqtr3d 2242	1 |- ( ph -> ( E ` T ) = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   _Vcvv 2776   C_ wss 3174   <.cop 3646   Fun wfun 5284   ` cfv 5290   NNcn 9071   ndxcnx 12944  Slot cslot 12946

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fv 5298  df-slot 12951

This theorem is referenced by:  strslss  12995