Theorem issubrg 14234

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 14232	. . 3 |- SubRing = ( r e. Ring |-> { s e. ~P ( Base ` r ) | ( ( r ↾s s ) e. Ring /\ ( 1r ` r ) e. s ) } )
2	1	mptrcl 5729	. 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 5639	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
5		issubrg.b	. . . . . . . 8 |- B = ( Base ` R )
6	4, 5	eqtr4di 2282	. . . . . . 7 |- ( r = R -> ( Base ` r ) = B )
7	6	pweqd 3657	. . . . . 6 |- ( r = R -> ~P ( Base ` r ) = ~P B )
8		oveq1 6024	. . . . . . . 8 |- ( r = R -> ( r ↾s s ) = ( R ↾s s ) )
9	8	eleq1d 2300	. . . . . . 7 |- ( r = R -> ( ( r ↾s s ) e. Ring <-> ( R ↾s s ) e. Ring ) )
10		fveq2 5639	. . . . . . . . 9 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
11		issubrg.i	. . . . . . . . 9 |- .1. = ( 1r ` R )
12	10, 11	eqtr4di 2282	. . . . . . . 8 |- ( r = R -> ( 1r ` r ) = .1. )
13	12	eleq1d 2300	. . . . . . 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 2797	. . . . 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 13140	. . . . . . . . 9 |- Base Fn _V
18		elex 2814	. . . . . . . . 9 |- ( R e. Ring -> R e. _V )
19		funfvex 5656	. . . . . . . . . 10 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
20	19	funfni 5432	. . . . . . . . 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 2318	. . . . . . 7 |- ( R e. Ring -> B e. _V )
23	22	pwexd 4271	. . . . . 6 |- ( R e. Ring -> ~P B e. _V )
24		rabexg 4233	. . . . . 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 5740	. . . 4 |- ( R e. Ring -> (SubRing ` R ) = { s e. ~P B | ( ( R ↾s s ) e. Ring /\ .1. e. s ) } )
27	26	eleq2d 2301	. . 3 |- ( R e. Ring -> ( A e. (SubRing ` R ) <-> A e. { s e. ~P B | ( ( R ↾s s ) e. Ring /\ .1. e. s ) } ) )
28		oveq2 6025	. . . . . . . 8 |- ( s = A -> ( R ↾s s ) = ( R ↾s A ) )
29	28	eleq1d 2300	. . . . . . 7 |- ( s = A -> ( ( R ↾s s ) e. Ring <-> ( R ↾s A ) e. Ring ) )
30		eleq2 2295	. . . . . . 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 2962	. . . . 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 4246	. . . . . 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 563	. . . . 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 711	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 1397   e. wcel 2202   {crab 2514   _Vcvv 2802   C_ wss 3200   ~Pcpw 3652   Fn wfn 5321   ` cfv 5326  (class class class)co 6017   Basecbs 13081   ↾_s cress 13082   1rcur 13971   Ringcrg 14008  SubRingcsubrg 14230

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6020  df-inn 9143  df-ndx 13084  df-slot 13085  df-base 13087  df-subrg 14232

This theorem is referenced by:  subrgss  14235  subrgid  14236  subrgring  14237  subrgrcl  14239  subrg1cl  14242  issubrg2  14254  subsubrg  14258  subrgpropd  14266