Theorem issubrg 13720

Description: The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Proof shortened by AV, 12-Oct-2020.)

Hypotheses

Ref	Expression
issubrg.b	|- B = ( Base ` R )
issubrg.i	|- .1. = ( 1r ` R )

Assertion

Ref	Expression
issubrg	|- ( A e. (SubRing ` R ) <-> ( ( R e. Ring /\ ( R ↾s A ) e. Ring ) /\ ( A C_ B /\ .1. e. A ) ) )

Proof of Theorem issubrg

Dummy variables s r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-subrg 13718	. . 3 |- SubRing = ( r e. Ring |-> { s e. ~P ( Base ` r ) | ( ( r ↾s s ) e. Ring /\ ( 1r ` r ) e. s ) } )
2	1	mptrcl 5641	. 2 |- ( A e. (SubRing ` R ) -> R e. Ring )
3		simpll 527	. 2 |- ( ( ( R e. Ring /\ ( R ↾s A ) e. Ring ) /\ ( A C_ B /\ .1. e. A ) ) -> R e. Ring )
4		fveq2 5555	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
5		issubrg.b	. . . . . . . 8 |- B = ( Base ` R )
6	4, 5	eqtr4di 2244	. . . . . . 7 |- ( r = R -> ( Base ` r ) = B )
7	6	pweqd 3607	. . . . . 6 |- ( r = R -> ~P ( Base ` r ) = ~P B )
8		oveq1 5926	. . . . . . . 8 |- ( r = R -> ( r ↾s s ) = ( R ↾s s ) )
9	8	eleq1d 2262	. . . . . . 7 |- ( r = R -> ( ( r ↾s s ) e. Ring <-> ( R ↾s s ) e. Ring ) )
10		fveq2 5555	. . . . . . . . 9 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
11		issubrg.i	. . . . . . . . 9 |- .1. = ( 1r ` R )
12	10, 11	eqtr4di 2244	. . . . . . . 8 |- ( r = R -> ( 1r ` r ) = .1. )
13	12	eleq1d 2262	. . . . . . 7 |- ( r = R -> ( ( 1r ` r ) e. s <-> .1. e. s ) )
14	9, 13	anbi12d 473	. . . . . 6 |- ( r = R -> ( ( ( r ↾s s ) e. Ring /\ ( 1r ` r ) e. s ) <-> ( ( R ↾s s ) e. Ring /\ .1. e. s ) ) )
15	7, 14	rabeqbidv 2755	. . . . 5 |- ( r = R -> { s e. ~P ( Base ` r ) | ( ( r ↾s s ) e. Ring /\ ( 1r ` r ) e. s ) } = { s e. ~P B | ( ( R ↾s s ) e. Ring /\ .1. e. s ) } )
16		id 19	. . . . 5 |- ( R e. Ring -> R e. Ring )
17		basfn 12679	. . . . . . . . 9 |- Base Fn _V
18		elex 2771	. . . . . . . . 9 |- ( R e. Ring -> R e. _V )
19		funfvex 5572	. . . . . . . . . 10 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
20	19	funfni 5355	. . . . . . . . 9 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
21	17, 18, 20	sylancr 414	. . . . . . . 8 |- ( R e. Ring -> ( Base ` R ) e. _V )
22	5, 21	eqeltrid 2280	. . . . . . 7 |- ( R e. Ring -> B e. _V )
23	22	pwexd 4211	. . . . . 6 |- ( R e. Ring -> ~P B e. _V )
24		rabexg 4173	. . . . . 6 |- ( ~P B e. _V -> { s e. ~P B | ( ( R ↾s s ) e. Ring /\ .1. e. s ) } e. _V )
25	23, 24	syl 14	. . . . 5 |- ( R e. Ring -> { s e. ~P B | ( ( R ↾s s ) e. Ring /\ .1. e. s ) } e. _V )
26	1, 15, 16, 25	fvmptd3 5652	. . . 4 |- ( R e. Ring -> (SubRing ` R ) = { s e. ~P B | ( ( R ↾s s ) e. Ring /\ .1. e. s ) } )
27	26	eleq2d 2263	. . 3 |- ( R e. Ring -> ( A e. (SubRing ` R ) <-> A e. { s e. ~P B | ( ( R ↾s s ) e. Ring /\ .1. e. s ) } ) )
28		oveq2 5927	. . . . . . . 8 |- ( s = A -> ( R ↾s s ) = ( R ↾s A ) )
29	28	eleq1d 2262	. . . . . . 7 |- ( s = A -> ( ( R ↾s s ) e. Ring <-> ( R ↾s A ) e. Ring ) )
30		eleq2 2257	. . . . . . 7 |- ( s = A -> ( .1. e. s <-> .1. e. A ) )
31	29, 30	anbi12d 473	. . . . . 6 |- ( s = A -> ( ( ( R ↾s s ) e. Ring /\ .1. e. s ) <-> ( ( R ↾s A ) e. Ring /\ .1. e. A ) ) )
32	31	elrab 2917	. . . . 5 |- ( A e. { s e. ~P B | ( ( R ↾s s ) e. Ring /\ .1. e. s ) } <-> ( A e. ~P B /\ ( ( R ↾s A ) e. Ring /\ .1. e. A ) ) )
33	32	a1i 9	. . . 4 |- ( R e. Ring -> ( A e. { s e. ~P B | ( ( R ↾s s ) e. Ring /\ .1. e. s ) } <-> ( A e. ~P B /\ ( ( R ↾s A ) e. Ring /\ .1. e. A ) ) ) )
34		elpw2g 4186	. . . . . 6 |- ( B e. _V -> ( A e. ~P B <-> A C_ B ) )
35	22, 34	syl 14	. . . . 5 |- ( R e. Ring -> ( A e. ~P B <-> A C_ B ) )
36	35	anbi1d 465	. . . 4 |- ( R e. Ring -> ( ( A e. ~P B /\ ( ( R ↾s A ) e. Ring /\ .1. e. A ) ) <-> ( A C_ B /\ ( ( R ↾s A ) e. Ring /\ .1. e. A ) ) ) )
37		an12 561	. . . . 5 |- ( ( A C_ B /\ ( ( R ↾s A ) e. Ring /\ .1. e. A ) ) <-> ( ( R ↾s A ) e. Ring /\ ( A C_ B /\ .1. e. A ) ) )
38	37	a1i 9	. . . 4 |- ( R e. Ring -> ( ( A C_ B /\ ( ( R ↾s A ) e. Ring /\ .1. e. A ) ) <-> ( ( R ↾s A ) e. Ring /\ ( A C_ B /\ .1. e. A ) ) ) )
39	33, 36, 38	3bitrd 214	. . 3 |- ( R e. Ring -> ( A e. { s e. ~P B | ( ( R ↾s s ) e. Ring /\ .1. e. s ) } <-> ( ( R ↾s A ) e. Ring /\ ( A C_ B /\ .1. e. A ) ) ) )
40		ibar 301	. . . 4 |- ( R e. Ring -> ( ( R ↾s A ) e. Ring <-> ( R e. Ring /\ ( R ↾s A ) e. Ring ) ) )
41	40	anbi1d 465	. . 3 |- ( R e. Ring -> ( ( ( R ↾s A ) e. Ring /\ ( A C_ B /\ .1. e. A ) ) <-> ( ( R e. Ring /\ ( R ↾s A ) e. Ring ) /\ ( A C_ B /\ .1. e. A ) ) ) )
42	27, 39, 41	3bitrd 214	. 2 |- ( R e. Ring -> ( A e. (SubRing ` R ) <-> ( ( R e. Ring /\ ( R ↾s A ) e. Ring ) /\ ( A C_ B /\ .1. e. A ) ) ) )
43	2, 3, 42	pm5.21nii 705	1 |- ( A e. (SubRing ` R ) <-> ( ( R e. Ring /\ ( R ↾s A ) e. Ring ) /\ ( A C_ B /\ .1. e. A ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   {crab 2476   _Vcvv 2760   C_ wss 3154   ~Pcpw 3602   Fn wfn 5250   ` cfv 5255  (class class class)co 5919   Basecbs 12621   ↾_s cress 12622   1rcur 13458   Ringcrg 13495  SubRingcsubrg 13716

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5922  df-inn 8985  df-ndx 12624  df-slot 12625  df-base 12627  df-subrg 13718

This theorem is referenced by:  subrgss  13721  subrgid  13722  subrgring  13723  subrgrcl  13725  subrg1cl  13728  issubrg2  13740  subsubrg  13744  subrgpropd  13752