Theorem strslfvd 12435

Description: Deduction version of strslfv 12438. (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 110	. . 3 |- E = Slot ( E ` ndx )
3		strfvd.s	. . 3 |- ( ph -> S e. V )
4	1	simpri 112	. . . 4 |- ( E ` ndx ) e. NN
5	4	a1i 9	. . 3 |- ( ph -> ( E ` ndx ) e. NN )
6	2, 3, 5	strnfvnd 12414	. 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 5526	. . 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 2199	1 |- ( ph -> C = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   <.cop 3579   Fun wfun 5182   ` cfv 5188   NNcn 8857   ndxcnx 12391  Slot cslot 12393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fv 5196  df-slot 12398

This theorem is referenced by:  strslssd  12440  1strbas  12494  2strbasg  12496  2stropg  12497