Theorem strslfv2d 13188

Description: Deduction version of strslfv 13190. (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 13165	. 2 |- ( ph -> ( E ` S ) = ( S ` ( E ` ndx ) ) )
7		cnvcnv2 5197	. . . 4 |- `' `' S = ( S |` _V )
8	7	fveq1i 5649	. . 3 |- ( `' `' S ` ( E ` ndx ) ) = ( ( S |` _V ) ` ( E ` ndx ) )
9	5	elexd 2817	. . . 4 |- ( ph -> ( E ` ndx ) e. _V )
10		fvres 5672	. . . 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 2276	. 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 2817	. . . . . 6 |- ( ph -> C e. _V )
17	9, 16	opelxpd 4764	. . . . 5 |- ( ph -> <. ( E ` ndx ) , C >. e. ( _V X. _V ) )
18	14, 17	elind 3394	. . . 4 |- ( ph -> <. ( E ` ndx ) , C >. e. ( S i^i ( _V X. _V ) ) )
19		cnvcnv 5196	. . . 4 |- `' `' S = ( S i^i ( _V X. _V ) )
20	18, 19	eleqtrrdi 2325	. . 3 |- ( ph -> <. ( E ` ndx ) , C >. e. `' `' S )
21		funopfv 5692	. . 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 2271	1 |- ( ph -> C = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _Vcvv 2803   i^i cin 3200   <.cop 3676   X. cxp 4729   `'ccnv 4730   |` cres 4733   Fun wfun 5327   ` cfv 5333   NNcn 9185   ndxcnx 13142  Slot cslot 13144

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fv 5341  df-slot 13149

This theorem is referenced by:  strslfv2  13189  strslfv  13190  strslfv3  13191  opelstrsl  13260