Theorem strslfv3 13191

Description: Variant on strslfv 13190 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 )
strslfv3.s	|- ( ph -> S Struct X )
strslfv3.e	|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
strslfv3.n	|- ( ph -> { <. ( 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		strslfv3.e	. . 3 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
3		strfv3.u	. . . 4 |- ( ph -> U = S )
4		strslfv3.s	. . . . 5 |- ( ph -> S Struct X )
5		structex 13157	. . . . 5 |- ( S Struct X -> S e. _V )
6	4, 5	syl 14	. . . 4 |- ( ph -> S e. _V )
7	3, 6	eqeltrd 2308	. . 3 |- ( ph -> U e. _V )
8		structfung 13162	. . . . 5 |- ( S Struct X -> Fun `' `' S )
9	4, 8	syl 14	. . . 4 |- ( ph -> Fun `' `' S )
10	3	cnveqd 4912	. . . . . 6 |- ( ph -> `' U = `' S )
11	10	cnveqd 4912	. . . . 5 |- ( ph -> `' `' U = `' `' S )
12	11	funeqd 5355	. . . 4 |- ( ph -> ( Fun `' `' U <-> Fun `' `' S ) )
13	9, 12	mpbird 167	. . 3 |- ( ph -> Fun `' `' U )
14		strslfv3.n	. . . . 5 |- ( ph -> { <. ( E ` ndx ) , C >. } C_ S )
15	2	simpri 113	. . . . . . 7 |- ( E ` ndx ) e. NN
16		strfv3.c	. . . . . . 7 |- ( ph -> C e. V )
17		opexg 4326	. . . . . . 7 |- ( ( ( E ` ndx ) e. NN /\ C e. V ) -> <. ( E ` ndx ) , C >. e. _V )
18	15, 16, 17	sylancr 414	. . . . . 6 |- ( ph -> <. ( E ` ndx ) , C >. e. _V )
19		snssg 3812	. . . . . 6 |- ( <. ( E ` ndx ) , C >. e. _V -> ( <. ( E ` ndx ) , C >. e. S <-> { <. ( E ` ndx ) , C >. } C_ S ) )
20	18, 19	syl 14	. . . . 5 |- ( ph -> ( <. ( E ` ndx ) , C >. e. S <-> { <. ( E ` ndx ) , C >. } C_ S ) )
21	14, 20	mpbird 167	. . . 4 |- ( ph -> <. ( E ` ndx ) , C >. e. S )
22	21, 3	eleqtrrd 2311	. . 3 |- ( ph -> <. ( E ` ndx ) , C >. e. U )
23	2, 7, 13, 22, 16	strslfv2d 13188	. 2 |- ( ph -> C = ( E ` U ) )
24	1, 23	eqtr4id 2283	1 |- ( ph -> A = C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   _Vcvv 2803   C_ wss 3201   {csn 3673   <.cop 3676   class class class wbr 4093   `'ccnv 4730   Fun wfun 5327   ` cfv 5333   NNcn 9185   Struct cstr 13141   ndxcnx 13142  Slot cslot 13144

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fv 5341  df-struct 13147  df-slot 13149

This theorem is referenced by:  prdsbaslemss  13420