Theorem strslfvd 13069

Description: Deduction version of strslfv 13072. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 30-Jan-2023.)

Hypotheses

Ref	Expression
strslfvd.e	|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
strfvd.s	|- ( ph -> S e. V )
strfvd.f	|- ( ph -> Fun S )
strfvd.n	|- ( ph -> <. ( E ` ndx ) , C >. e. S )

Assertion

Ref	Expression
strslfvd	|- ( ph -> C = ( E ` S ) )

Proof of Theorem strslfvd

Step	Hyp	Ref	Expression
1		strslfvd.e	. . . 4 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
2	1	simpli 111	. . 3 |- E = Slot ( E ` ndx )
3		strfvd.s	. . 3 |- ( ph -> S e. V )
4	1	simpri 113	. . . 4 |- ( E ` ndx ) e. NN
5	4	a1i 9	. . 3 |- ( ph -> ( E ` ndx ) e. NN )
6	2, 3, 5	strnfvnd 13047	. 2 |- ( ph -> ( E ` S ) = ( S ` ( E ` ndx ) ) )
7		strfvd.f	. . 3 |- ( ph -> Fun S )
8		strfvd.n	. . 3 |- ( ph -> <. ( E ` ndx ) , C >. e. S )
9		funopfv 5670	. . 3 |- ( Fun S -> ( <. ( E ` ndx ) , C >. e. S -> ( S ` ( E ` ndx ) ) = C ) )
10	7, 8, 9	sylc 62	. 2 |- ( ph -> ( S ` ( E ` ndx ) ) = C )
11	6, 10	eqtr2d 2263	1 |- ( ph -> C = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   <.cop 3669   Fun wfun 5311   ` cfv 5317   NNcn 9106   ndxcnx 13024  Slot cslot 13026

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 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

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 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-iota 5277  df-fun 5319  df-fv 5325  df-slot 13031

This theorem is referenced by:  strslssd  13074  1strbas  13145  2strbasg  13148  2stropg  13149