Theorem strslfvd 12949

Description: Deduction version of strslfv 12952. (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 12927	. 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 5631	. . 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 2240	1 |- ( ph -> C = ( E ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   <.cop 3641   Fun wfun 5274   ` cfv 5280   NNcn 9056   ndxcnx 12904  Slot cslot 12906

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 4170  ax-pow 4226  ax-pr 4261  ax-un 4488

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 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-iota 5241  df-fun 5282  df-fv 5288  df-slot 12911

This theorem is referenced by:  strslssd  12954  1strbas  13024  2strbasg  13027  2stropg  13028