Theorem strslssd 13079

Description: Deduction version of strslss 13080. (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 3225	. . 3 |- ( ph -> <. ( E ` ndx ) , C >. e. T )
7	1, 2, 3, 6	strslfvd 13074	. 2 |- ( ph -> C = ( E ` T ) )
8	2, 4	ssexd 4224	. . 3 |- ( ph -> S e. _V )
9		funss 5337	. . . 4 |- ( S C_ T -> ( Fun T -> Fun S ) )
10	4, 3, 9	sylc 62	. . 3 |- ( ph -> Fun S )
11	1, 8, 10, 5	strslfvd 13074	. 2 |- ( ph -> C = ( E ` S ) )
12	7, 11	eqtr3d 2264	1 |- ( ph -> ( E ` T ) = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799   C_ wss 3197   <.cop 3669   Fun wfun 5312   ` cfv 5318   NNcn 9110   ndxcnx 13029  Slot cslot 13031

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fv 5326  df-slot 13036

This theorem is referenced by:  strslss  13080