Theorem strslfv3 12667

Description: Variant on strslfv 12666 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.) (Revised by Jim Kingdon, 30-Jan-2023.)

Hypotheses

Ref	Expression
strfv3.u	|- ( ph -> U = S )
strfv3.s	|- S Struct X
strslfv3.e	|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
strfv3.n	|- { <. ( E ` ndx ) , C >. } C_ S
strfv3.c	|- ( ph -> C e. V )
strfv3.a	|- A = ( E ` U )

Assertion

Ref	Expression
strslfv3	|- ( ph -> A = C )

Proof of Theorem strslfv3

Step	Hyp	Ref	Expression
1		strfv3.a	. 2 |- A = ( E ` U )
2		strfv3.c	. . . 4 |- ( ph -> C e. V )
3		strfv3.s	. . . . 5 |- S Struct X
4		strslfv3.e	. . . . 5 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
5		strfv3.n	. . . . 5 |- { <. ( E ` ndx ) , C >. } C_ S
6	3, 4, 5	strslfv 12666	. . . 4 |- ( C e. V -> C = ( E ` S ) )
7	2, 6	syl 14	. . 3 |- ( ph -> C = ( E ` S ) )
8		strfv3.u	. . . 4 |- ( ph -> U = S )
9	8	fveq2d 5559	. . 3 |- ( ph -> ( E ` U ) = ( E ` S ) )
10	7, 9	eqtr4d 2229	. 2 |- ( ph -> C = ( E ` U ) )
11	1, 10	eqtr4id 2245	1 |- ( ph -> A = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   C_ wss 3154   {csn 3619   <.cop 3622   class class class wbr 4030   ` cfv 5255   NNcn 8984   Struct cstr 12617   ndxcnx 12618  Slot cslot 12620

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-in1 615  ax-in2 616  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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fv 5263  df-struct 12623  df-slot 12625

This theorem is referenced by: (None)