Theorem strslfvd 12565

Description: Deduction version of strslfv 12568. (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 111	. . 3 |- E = Slot ( E ` ndx )
3		strfvd.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		strfvd.f	. . 3 |- ( ph -> Fun S )
8		strfvd.n	. . 3 |- ( ph -> <. ( E ` ndx ) , C >. e. S )
9		funopfv 5579	. . 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 2223	1 |- ( ph -> C = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   <.cop 3613   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-iota 5199  df-fun 5240  df-fv 5246  df-slot 12527

This theorem is referenced by:  strslssd  12570  1strbas  12640  2strbasg  12642  2stropg  12643