Theorem issubrng 13695

Description: The subring of non-unital ring predicate. (Contributed by AV, 14-Feb-2025.)

Hypothesis

Ref	Expression
issubrng.b	|- B = ( Base ` R )

Assertion

Ref	Expression
issubrng	|- ( A e. (SubRng ` R ) <-> ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) )

Proof of Theorem issubrng

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

Step	Hyp	Ref	Expression
1		df-subrng 13694	. . 3 |- SubRng = ( w e. Rng |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Rng } )
2	1	mptrcl 5640	. 2 |- ( A e. (SubRng ` R ) -> R e. Rng )
3		simp1 999	. 2 |- ( ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) -> R e. Rng )
4		df-subrng 13694	. . . . 5 |- SubRng = ( r e. Rng |-> { s e. ~P ( Base ` r ) | ( r ↾s s ) e. Rng } )
5		fveq2 5554	. . . . . . 7 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
6	5	pweqd 3606	. . . . . 6 |- ( r = R -> ~P ( Base ` r ) = ~P ( Base ` R ) )
7		oveq1 5925	. . . . . . 7 |- ( r = R -> ( r ↾s s ) = ( R ↾s s ) )
8	7	eleq1d 2262	. . . . . 6 |- ( r = R -> ( ( r ↾s s ) e. Rng <-> ( R ↾s s ) e. Rng ) )
9	6, 8	rabeqbidv 2755	. . . . 5 |- ( r = R -> { s e. ~P ( Base ` r ) | ( r ↾s s ) e. Rng } = { s e. ~P ( Base ` R ) | ( R ↾s s ) e. Rng } )
10		id 19	. . . . 5 |- ( R e. Rng -> R e. Rng )
11		basfn 12676	. . . . . . . 8 |- Base Fn _V
12		elex 2771	. . . . . . . 8 |- ( R e. Rng -> R e. _V )
13		funfvex 5571	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
14	13	funfni 5354	. . . . . . . 8 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
15	11, 12, 14	sylancr 414	. . . . . . 7 |- ( R e. Rng -> ( Base ` R ) e. _V )
16	15	pwexd 4210	. . . . . 6 |- ( R e. Rng -> ~P ( Base ` R ) e. _V )
17		rabexg 4172	. . . . . 6 |- ( ~P ( Base ` R ) e. _V -> { s e. ~P ( Base ` R ) | ( R ↾s s ) e. Rng } e. _V )
18	16, 17	syl 14	. . . . 5 |- ( R e. Rng -> { s e. ~P ( Base ` R ) | ( R ↾s s ) e. Rng } e. _V )
19	4, 9, 10, 18	fvmptd3 5651	. . . 4 |- ( R e. Rng -> (SubRng ` R ) = { s e. ~P ( Base ` R ) | ( R ↾s s ) e. Rng } )
20	19	eleq2d 2263	. . 3 |- ( R e. Rng -> ( A e. (SubRng ` R ) <-> A e. { s e. ~P ( Base ` R ) | ( R ↾s s ) e. Rng } ) )
21		oveq2 5926	. . . . . 6 |- ( s = A -> ( R ↾s s ) = ( R ↾s A ) )
22	21	eleq1d 2262	. . . . 5 |- ( s = A -> ( ( R ↾s s ) e. Rng <-> ( R ↾s A ) e. Rng ) )
23	22	elrab 2916	. . . 4 |- ( A e. { s e. ~P ( Base ` R ) | ( R ↾s s ) e. Rng } <-> ( A e. ~P ( Base ` R ) /\ ( R ↾s A ) e. Rng ) )
24		issubrng.b	. . . . . . . . 9 |- B = ( Base ` R )
25	24	eqcomi 2197	. . . . . . . 8 |- ( Base ` R ) = B
26	25	sseq2i 3206	. . . . . . 7 |- ( A C_ ( Base ` R ) <-> A C_ B )
27	26	anbi2i 457	. . . . . 6 |- ( ( ( R ↾s A ) e. Rng /\ A C_ ( Base ` R ) ) <-> ( ( R ↾s A ) e. Rng /\ A C_ B ) )
28		ibar 301	. . . . . 6 |- ( R e. Rng -> ( ( ( R ↾s A ) e. Rng /\ A C_ B ) <-> ( R e. Rng /\ ( ( R ↾s A ) e. Rng /\ A C_ B ) ) ) )
29	27, 28	bitrid 192	. . . . 5 |- ( R e. Rng -> ( ( ( R ↾s A ) e. Rng /\ A C_ ( Base ` R ) ) <-> ( R e. Rng /\ ( ( R ↾s A ) e. Rng /\ A C_ B ) ) ) )
30		ancom 266	. . . . . 6 |- ( ( A e. ~P ( Base ` R ) /\ ( R ↾s A ) e. Rng ) <-> ( ( R ↾s A ) e. Rng /\ A e. ~P ( Base ` R ) ) )
31		elpw2g 4185	. . . . . . . 8 |- ( ( Base ` R ) e. _V -> ( A e. ~P ( Base ` R ) <-> A C_ ( Base ` R ) ) )
32	15, 31	syl 14	. . . . . . 7 |- ( R e. Rng -> ( A e. ~P ( Base ` R ) <-> A C_ ( Base ` R ) ) )
33	32	anbi2d 464	. . . . . 6 |- ( R e. Rng -> ( ( ( R ↾s A ) e. Rng /\ A e. ~P ( Base ` R ) ) <-> ( ( R ↾s A ) e. Rng /\ A C_ ( Base ` R ) ) ) )
34	30, 33	bitrid 192	. . . . 5 |- ( R e. Rng -> ( ( A e. ~P ( Base ` R ) /\ ( R ↾s A ) e. Rng ) <-> ( ( R ↾s A ) e. Rng /\ A C_ ( Base ` R ) ) ) )
35		3anass 984	. . . . . 6 |- ( ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) <-> ( R e. Rng /\ ( ( R ↾s A ) e. Rng /\ A C_ B ) ) )
36	35	a1i 9	. . . . 5 |- ( R e. Rng -> ( ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) <-> ( R e. Rng /\ ( ( R ↾s A ) e. Rng /\ A C_ B ) ) ) )
37	29, 34, 36	3bitr4d 220	. . . 4 |- ( R e. Rng -> ( ( A e. ~P ( Base ` R ) /\ ( R ↾s A ) e. Rng ) <-> ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) ) )
38	23, 37	bitrid 192	. . 3 |- ( R e. Rng -> ( A e. { s e. ~P ( Base ` R ) | ( R ↾s s ) e. Rng } <-> ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) ) )
39	20, 38	bitrd 188	. 2 |- ( R e. Rng -> ( A e. (SubRng ` R ) <-> ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) ) )
40	2, 3, 39	pm5.21nii 705	1 |- ( A e. (SubRng ` R ) <-> ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   {crab 2476   _Vcvv 2760   C_ wss 3153   ~Pcpw 3601   Fn wfn 5249   ` cfv 5254  (class class class)co 5918   Basecbs 12618   ↾_s cress 12619  Rngcrng 13428  SubRngcsubrng 13693

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 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

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 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-ov 5921  df-inn 8983  df-ndx 12621  df-slot 12622  df-base 12624  df-subrng 13694

This theorem is referenced by:  subrngss  13696  subrngid  13697  subrngrng  13698  subrngrcl  13699  issubrng2  13706  subsubrng  13710  subrngpropd  13712  rng2idlsubrng  14013