Theorem issubrg 14098

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 14096	. . 3 |- SubRing = ( r e. Ring |-> { s e. ~P ( Base ` r ) | ( ( r ↾s s ) e. Ring /\ ( 1r ` r ) e. s ) } )
2	1	mptrcl 5685	. 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 5599	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
5		issubrg.b	. . . . . . . 8 |- B = ( Base ` R )
6	4, 5	eqtr4di 2258	. . . . . . 7 |- ( r = R -> ( Base ` r ) = B )
7	6	pweqd 3631	. . . . . 6 |- ( r = R -> ~P ( Base ` r ) = ~P B )
8		oveq1 5974	. . . . . . . 8 |- ( r = R -> ( r ↾s s ) = ( R ↾s s ) )
9	8	eleq1d 2276	. . . . . . 7 |- ( r = R -> ( ( r ↾s s ) e. Ring <-> ( R ↾s s ) e. Ring ) )
10		fveq2 5599	. . . . . . . . 9 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
11		issubrg.i	. . . . . . . . 9 |- .1. = ( 1r ` R )
12	10, 11	eqtr4di 2258	. . . . . . . 8 |- ( r = R -> ( 1r ` r ) = .1. )
13	12	eleq1d 2276	. . . . . . 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 2771	. . . . 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 13005	. . . . . . . . 9 |- Base Fn _V
18		elex 2788	. . . . . . . . 9 |- ( R e. Ring -> R e. _V )
19		funfvex 5616	. . . . . . . . . 10 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
20	19	funfni 5395	. . . . . . . . 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 2294	. . . . . . 7 |- ( R e. Ring -> B e. _V )
23	22	pwexd 4241	. . . . . 6 |- ( R e. Ring -> ~P B e. _V )
24		rabexg 4203	. . . . . 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 5696	. . . 4 |- ( R e. Ring -> (SubRing ` R ) = { s e. ~P B | ( ( R ↾s s ) e. Ring /\ .1. e. s ) } )
27	26	eleq2d 2277	. . 3 |- ( R e. Ring -> ( A e. (SubRing ` R ) <-> A e. { s e. ~P B | ( ( R ↾s s ) e. Ring /\ .1. e. s ) } ) )
28		oveq2 5975	. . . . . . . 8 |- ( s = A -> ( R ↾s s ) = ( R ↾s A ) )
29	28	eleq1d 2276	. . . . . . 7 |- ( s = A -> ( ( R ↾s s ) e. Ring <-> ( R ↾s A ) e. Ring ) )
30		eleq2 2271	. . . . . . 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 2936	. . . . 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 4216	. . . . . 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 706	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 1373   e. wcel 2178   {crab 2490   _Vcvv 2776   C_ wss 3174   ~Pcpw 3626   Fn wfn 5285   ` cfv 5290  (class class class)co 5967   Basecbs 12947   ↾_s cress 12948   1rcur 13836   Ringcrg 13873  SubRingcsubrg 14094

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-ov 5970  df-inn 9072  df-ndx 12950  df-slot 12951  df-base 12953  df-subrg 14096

This theorem is referenced by:  subrgss  14099  subrgid  14100  subrgring  14101  subrgrcl  14103  subrg1cl  14106  issubrg2  14118  subsubrg  14122  subrgpropd  14130