Theorem strslfv2d 12925

Description: Deduction version of strslfv 12927. (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 12902	. 2 |- ( ph -> ( E ` S ) = ( S ` ( E ` ndx ) ) )
7		cnvcnv2 5142	. . . 4 |- `' `' S = ( S |` _V )
8	7	fveq1i 5587	. . 3 |- ( `' `' S ` ( E ` ndx ) ) = ( ( S |` _V ) ` ( E ` ndx ) )
9	5	elexd 2787	. . . 4 |- ( ph -> ( E ` ndx ) e. _V )
10		fvres 5610	. . . 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 2251	. 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 2787	. . . . . 6 |- ( ph -> C e. _V )
17	9, 16	opelxpd 4713	. . . . 5 |- ( ph -> <. ( E ` ndx ) , C >. e. ( _V X. _V ) )
18	14, 17	elind 3360	. . . 4 |- ( ph -> <. ( E ` ndx ) , C >. e. ( S i^i ( _V X. _V ) ) )
19		cnvcnv 5141	. . . 4 |- `' `' S = ( S i^i ( _V X. _V ) )
20	18, 19	eleqtrrdi 2300	. . 3 |- ( ph -> <. ( E ` ndx ) , C >. e. `' `' S )
21		funopfv 5628	. . 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 2246	1 |- ( ph -> C = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   _Vcvv 2773   i^i cin 3167   <.cop 3638   X. cxp 4678   `'ccnv 4679   |` cres 4682   Fun wfun 5271   ` cfv 5277   NNcn 9049   ndxcnx 12879  Slot cslot 12881

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258  ax-un 4485

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3001  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-br 4049  df-opab 4111  df-mpt 4112  df-id 4345  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-iota 5238  df-fun 5279  df-fv 5285  df-slot 12886

This theorem is referenced by:  strslfv2  12926  strslfv  12927  strslfv3  12928  opelstrsl  12996