Theorem strslfv3 13376

Description: Variant on strslfv 13375 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 13342	. . . . 5 |- ( S Struct X -> S e. _V )
6	4, 5	syl 14	. . . 4 |- ( ph -> S e. _V )
7	3, 6	eqeltrd 2315	. . 3 |- ( ph -> U e. _V )
8		structfung 13347	. . . . 5 |- ( S Struct X -> Fun `' `' S )
9	4, 8	syl 14	. . . 4 |- ( ph -> Fun `' `' S )
10	3	cnveqd 4951	. . . . . 6 |- ( ph -> `' U = `' S )
11	10	cnveqd 4951	. . . . 5 |- ( ph -> `' `' U = `' `' S )
12	11	funeqd 5394	. . . 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 4363	. . . . . . 7 |- ( ( ( E ` ndx ) e. NN /\ C e. V ) -> <. ( E ` ndx ) , C >. e. _V )
18	15, 16, 17	sylancr 418	. . . . . 6 |- ( ph -> <. ( E ` ndx ) , C >. e. _V )
19		snssg 3844	. . . . . 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 2318	. . 3 |- ( ph -> <. ( E ` ndx ) , C >. e. U )
23	2, 7, 13, 22, 16	strslfv2d 13373	. 2 |- ( ph -> C = ( E ` U ) )
24	1, 23	eqtr4id 2290	1 |- ( ph -> A = C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   {csn 3705   <.cop 3708   class class class wbr 4125   `'ccnv 4768   Fun wfun 5366   ` cfv 5372   NNcn 9283   Struct cstr 13326   ndxcnx 13327  Slot cslot 13329

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fv 5380  df-struct 13332  df-slot 13334

This theorem is referenced by:  prdsbaslemss  14151