Theorem issubrg 13777

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 13775	. . 3 |- SubRing = ( r e. Ring |-> { s e. ~P ( Base ` r ) | ( ( r ↾s s ) e. Ring /\ ( 1r ` r ) e. s ) } )
2	1	mptrcl 5644	. 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 5558	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
5		issubrg.b	. . . . . . . 8 |- B = ( Base ` R )
6	4, 5	eqtr4di 2247	. . . . . . 7 |- ( r = R -> ( Base ` r ) = B )
7	6	pweqd 3610	. . . . . 6 |- ( r = R -> ~P ( Base ` r ) = ~P B )
8		oveq1 5929	. . . . . . . 8 |- ( r = R -> ( r ↾s s ) = ( R ↾s s ) )
9	8	eleq1d 2265	. . . . . . 7 |- ( r = R -> ( ( r ↾s s ) e. Ring <-> ( R ↾s s ) e. Ring ) )
10		fveq2 5558	. . . . . . . . 9 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
11		issubrg.i	. . . . . . . . 9 |- .1. = ( 1r ` R )
12	10, 11	eqtr4di 2247	. . . . . . . 8 |- ( r = R -> ( 1r ` r ) = .1. )
13	12	eleq1d 2265	. . . . . . 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 2758	. . . . 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 12736	. . . . . . . . 9 |- Base Fn _V
18		elex 2774	. . . . . . . . 9 |- ( R e. Ring -> R e. _V )
19		funfvex 5575	. . . . . . . . . 10 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
20	19	funfni 5358	. . . . . . . . 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 2283	. . . . . . 7 |- ( R e. Ring -> B e. _V )
23	22	pwexd 4214	. . . . . 6 |- ( R e. Ring -> ~P B e. _V )
24		rabexg 4176	. . . . . 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 5655	. . . 4 |- ( R e. Ring -> (SubRing ` R ) = { s e. ~P B | ( ( R ↾s s ) e. Ring /\ .1. e. s ) } )
27	26	eleq2d 2266	. . 3 |- ( R e. Ring -> ( A e. (SubRing ` R ) <-> A e. { s e. ~P B | ( ( R ↾s s ) e. Ring /\ .1. e. s ) } ) )
28		oveq2 5930	. . . . . . . 8 |- ( s = A -> ( R ↾s s ) = ( R ↾s A ) )
29	28	eleq1d 2265	. . . . . . 7 |- ( s = A -> ( ( R ↾s s ) e. Ring <-> ( R ↾s A ) e. Ring ) )
30		eleq2 2260	. . . . . . 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 2920	. . . . 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 4189	. . . . . 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 2167   {crab 2479   _Vcvv 2763   C_ wss 3157   ~Pcpw 3605   Fn wfn 5253   ` cfv 5258  (class class class)co 5922   Basecbs 12678   ↾_s cress 12679   1rcur 13515   Ringcrg 13552  SubRingcsubrg 13773

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925  df-inn 8991  df-ndx 12681  df-slot 12682  df-base 12684  df-subrg 13775

This theorem is referenced by:  subrgss  13778  subrgid  13779  subrgring  13780  subrgrcl  13782  subrg1cl  13785  issubrg2  13797  subsubrg  13801  subrgpropd  13809