Theorem strslfv2d 13061

Description: Deduction version of strslfv 13063. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)

Hypotheses

Ref	Expression
strslfv2d.e	|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
strfv2d.s	|- ( ph -> S e. V )
strfv2d.f	|- ( ph -> Fun `' `' S )
strfv2d.n	|- ( ph -> <. ( E ` ndx ) , C >. e. S )
strfv2d.c	|- ( ph -> C e. W )

Assertion

Ref	Expression
strslfv2d	|- ( ph -> C = ( E ` S ) )

Proof of Theorem strslfv2d

Step	Hyp	Ref	Expression
1		strslfv2d.e	. . . 4 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
2	1	simpli 111	. . 3 |- E = Slot ( E ` ndx )
3		strfv2d.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 13038	. 2 |- ( ph -> ( E ` S ) = ( S ` ( E ` ndx ) ) )
7		cnvcnv2 5178	. . . 4 |- `' `' S = ( S |` _V )
8	7	fveq1i 5624	. . 3 |- ( `' `' S ` ( E ` ndx ) ) = ( ( S |` _V ) ` ( E ` ndx ) )
9	5	elexd 2813	. . . 4 |- ( ph -> ( E ` ndx ) e. _V )
10		fvres 5647	. . . 4 |- ( ( E ` ndx ) e. _V -> ( ( S |` _V ) ` ( E ` ndx ) ) = ( S ` ( E ` ndx ) ) )
11	9, 10	syl 14	. . 3 |- ( ph -> ( ( S |` _V ) ` ( E ` ndx ) ) = ( S ` ( E ` ndx ) ) )
12	8, 11	eqtrid 2274	. 2 |- ( ph -> ( `' `' S ` ( E ` ndx ) ) = ( S ` ( E ` ndx ) ) )
13		strfv2d.f	. . 3 |- ( ph -> Fun `' `' S )
14		strfv2d.n	. . . . 5 |- ( ph -> <. ( E ` ndx ) , C >. e. S )
15		strfv2d.c	. . . . . . 7 |- ( ph -> C e. W )
16	15	elexd 2813	. . . . . 6 |- ( ph -> C e. _V )
17	9, 16	opelxpd 4749	. . . . 5 |- ( ph -> <. ( E ` ndx ) , C >. e. ( _V X. _V ) )
18	14, 17	elind 3389	. . . 4 |- ( ph -> <. ( E ` ndx ) , C >. e. ( S i^i ( _V X. _V ) ) )
19		cnvcnv 5177	. . . 4 |- `' `' S = ( S i^i ( _V X. _V ) )
20	18, 19	eleqtrrdi 2323	. . 3 |- ( ph -> <. ( E ` ndx ) , C >. e. `' `' S )
21		funopfv 5665	. . 3 |- ( Fun `' `' S -> ( <. ( E ` ndx ) , C >. e. `' `' S -> ( `' `' S ` ( E ` ndx ) ) = C ) )
22	13, 20, 21	sylc 62	. 2 |- ( ph -> ( `' `' S ` ( E ` ndx ) ) = C )
23	6, 12, 22	3eqtr2rd 2269	1 |- ( ph -> C = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799   i^i cin 3196   <.cop 3669   X. cxp 4714   `'ccnv 4715   |` cres 4718   Fun wfun 5308   ` cfv 5314   NNcn 9098   ndxcnx 13015  Slot cslot 13017

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 4521

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 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-iota 5274  df-fun 5316  df-fv 5322  df-slot 13022

This theorem is referenced by:  strslfv2  13062  strslfv  13063  strslfv3  13064  opelstrsl  13133