Theorem strslfvd 12506

Description: Deduction version of strslfv 12509. (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 12484	. 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 5557	. . 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 2211	1 |- ( ph -> C = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   <.cop 3597   Fun wfun 5212   ` cfv 5218   NNcn 8921   ndxcnx 12461  Slot cslot 12463

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fv 5226  df-slot 12468

This theorem is referenced by:  strslssd  12511  1strbas  12578  2strbasg  12580  2stropg  12581