Theorem strslfv2d 12566

Description: Deduction version of strslfv 12568. (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 12543	. 2 |- ( ph -> ( E ` S ) = ( S ` ( E ` ndx ) ) )
7		cnvcnv2 5103	. . . 4 |- `' `' S = ( S |` _V )
8	7	fveq1i 5538	. . 3 |- ( `' `' S ` ( E ` ndx ) ) = ( ( S |` _V ) ` ( E ` ndx ) )
9	5	elexd 2765	. . . 4 |- ( ph -> ( E ` ndx ) e. _V )
10		fvres 5561	. . . 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 2234	. 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 2765	. . . . . 6 |- ( ph -> C e. _V )
17	9, 16	opelxpd 4680	. . . . 5 |- ( ph -> <. ( E ` ndx ) , C >. e. ( _V X. _V ) )
18	14, 17	elind 3335	. . . 4 |- ( ph -> <. ( E ` ndx ) , C >. e. ( S i^i ( _V X. _V ) ) )
19		cnvcnv 5102	. . . 4 |- `' `' S = ( S i^i ( _V X. _V ) )
20	18, 19	eleqtrrdi 2283	. . 3 |- ( ph -> <. ( E ` ndx ) , C >. e. `' `' S )
21		funopfv 5579	. . 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 2229	1 |- ( ph -> C = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   _Vcvv 2752   i^i cin 3143   <.cop 3613   X. cxp 4645   `'ccnv 4646   |` cres 4649   Fun wfun 5232   ` cfv 5238   NNcn 8954   ndxcnx 12520  Slot cslot 12522

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-iota 5199  df-fun 5240  df-fv 5246  df-slot 12527

This theorem is referenced by:  strslfv2  12567  strslfv  12568  opelstrsl  12637