Theorem strslfv3 12797

Description: Variant on strslfv 12796 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 12763	. . . . 5 |- ( S Struct X -> S e. _V )
6	4, 5	syl 14	. . . 4 |- ( ph -> S e. _V )
7	3, 6	eqeltrd 2281	. . 3 |- ( ph -> U e. _V )
8		structfung 12768	. . . . 5 |- ( S Struct X -> Fun `' `' S )
9	4, 8	syl 14	. . . 4 |- ( ph -> Fun `' `' S )
10	3	cnveqd 4852	. . . . . 6 |- ( ph -> `' U = `' S )
11	10	cnveqd 4852	. . . . 5 |- ( ph -> `' `' U = `' `' S )
12	11	funeqd 5290	. . . 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 4271	. . . . . . 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 3766	. . . . . 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 2284	. . 3 |- ( ph -> <. ( E ` ndx ) , C >. e. U )
23	2, 7, 13, 22, 16	strslfv2d 12794	. 2 |- ( ph -> C = ( E ` U ) )
24	1, 23	eqtr4id 2256	1 |- ( ph -> A = C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   e. wcel 2175   _Vcvv 2771   C_ wss 3165   {csn 3632   <.cop 3635   class class class wbr 4043   `'ccnv 4672   Fun wfun 5262   ` cfv 5268   NNcn 9018   Struct cstr 12747   ndxcnx 12748  Slot cslot 12750

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-iota 5229  df-fun 5270  df-fv 5276  df-struct 12753  df-slot 12755

This theorem is referenced by:  prdsbaslemss  13024