Theorem strslfv3 12724

Description: Variant on strslfv 12723 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 12723	. . . 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 5562	. . 3 |- ( ph -> ( E ` U ) = ( E ` S ) )
10	7, 9	eqtr4d 2232	. 2 |- ( ph -> C = ( E ` U ) )
11	1, 10	eqtr4id 2248	1 |- ( ph -> A = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   C_ wss 3157   {csn 3622   <.cop 3625   class class class wbr 4033   ` cfv 5258   NNcn 8990   Struct cstr 12674   ndxcnx 12675  Slot cslot 12677

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fv 5266  df-struct 12680  df-slot 12682

This theorem is referenced by: (None)