Theorem strslfv3 12007

Description: Variant on strslfv 12006 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.c	. . . 4 |- ( ph -> C e. V )
2		strfv3.s	. . . . 5 |- S Struct X
3		strslfv3.e	. . . . 5 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
4		strfv3.n	. . . . 5 |- { <. ( E ` ndx ) , C >. } C_ S
5	2, 3, 4	strslfv 12006	. . . 4 |- ( C e. V -> C = ( E ` S ) )
6	1, 5	syl 14	. . 3 |- ( ph -> C = ( E ` S ) )
7		strfv3.u	. . . 4 |- ( ph -> U = S )
8	7	fveq2d 5425	. . 3 |- ( ph -> ( E ` U ) = ( E ` S ) )
9	6, 8	eqtr4d 2175	. 2 |- ( ph -> C = ( E ` U ) )
10		strfv3.a	. 2 |- A = ( E ` U )
11	9, 10	syl6reqr 2191	1 |- ( ph -> A = C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   C_ wss 3071   {csn 3527   <.cop 3530   class class class wbr 3929   ` cfv 5123   NNcn 8723   Struct cstr 11958   ndxcnx 11959  Slot cslot 11961

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fv 5131  df-struct 11964  df-slot 11966

This theorem is referenced by: (None)