Theorem strslfv3 12439

Description: Variant on strslfv 12438 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 12438	. . . 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 5490	. . 3 |- ( ph -> ( E ` U ) = ( E ` S ) )
10	7, 9	eqtr4d 2201	. 2 |- ( ph -> C = ( E ` U ) )
11	1, 10	eqtr4id 2218	1 |- ( ph -> A = C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   C_ wss 3116   {csn 3576   <.cop 3579   class class class wbr 3982   ` cfv 5188   NNcn 8857   Struct cstr 12390   ndxcnx 12391  Slot cslot 12393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fv 5196  df-struct 12396  df-slot 12398

This theorem is referenced by: (None)