Theorem strslfv3 12461

Description: Variant on strslfv 12460 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 12460	. . . 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 5500	. . 3 |- ( ph -> ( E ` U ) = ( E ` S ) )
10	7, 9	eqtr4d 2206	. 2 |- ( ph -> C = ( E ` U ) )
11	1, 10	eqtr4id 2222	1 |- ( ph -> A = C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   C_ wss 3121   {csn 3583   <.cop 3586   class class class wbr 3989   ` cfv 5198   NNcn 8878   Struct cstr 12412   ndxcnx 12413  Slot cslot 12415

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fv 5206  df-struct 12418  df-slot 12420

This theorem is referenced by: (None)