Theorem issubrng 14203

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 14202	. . 3 |- SubRng = ( w e. Rng |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Rng } )
2	1	mptrcl 5725	. 2 |- ( A e. (SubRng ` R ) -> R e. Rng )
3		simp1 1021	. 2 |- ( ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) -> R e. Rng )
4		df-subrng 14202	. . . . 5 |- SubRng = ( r e. Rng |-> { s e. ~P ( Base ` r ) | ( r ↾s s ) e. Rng } )
5		fveq2 5635	. . . . . . 7 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
6	5	pweqd 3655	. . . . . 6 |- ( r = R -> ~P ( Base ` r ) = ~P ( Base ` R ) )
7		oveq1 6020	. . . . . . 7 |- ( r = R -> ( r ↾s s ) = ( R ↾s s ) )
8	7	eleq1d 2298	. . . . . 6 |- ( r = R -> ( ( r ↾s s ) e. Rng <-> ( R ↾s s ) e. Rng ) )
9	6, 8	rabeqbidv 2795	. . . . 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 13131	. . . . . . . 8 |- Base Fn _V
12		elex 2812	. . . . . . . 8 |- ( R e. Rng -> R e. _V )
13		funfvex 5652	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
14	13	funfni 5429	. . . . . . . 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 4269	. . . . . 6 |- ( R e. Rng -> ~P ( Base ` R ) e. _V )
17		rabexg 4231	. . . . . 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 5736	. . . 4 |- ( R e. Rng -> (SubRng ` R ) = { s e. ~P ( Base ` R ) | ( R ↾s s ) e. Rng } )
20	19	eleq2d 2299	. . 3 |- ( R e. Rng -> ( A e. (SubRng ` R ) <-> A e. { s e. ~P ( Base ` R ) | ( R ↾s s ) e. Rng } ) )
21		oveq2 6021	. . . . . 6 |- ( s = A -> ( R ↾s s ) = ( R ↾s A ) )
22	21	eleq1d 2298	. . . . 5 |- ( s = A -> ( ( R ↾s s ) e. Rng <-> ( R ↾s A ) e. Rng ) )
23	22	elrab 2960	. . . 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 2233	. . . . . . . 8 |- ( Base ` R ) = B
26	25	sseq2i 3252	. . . . . . 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 4244	. . . . . . . 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 1006	. . . . . 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 709	1 |- ( A e. (SubRng ` R ) <-> ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   {crab 2512   _Vcvv 2800   C_ wss 3198   ~Pcpw 3650   Fn wfn 5319   ` cfv 5324  (class class class)co 6013   Basecbs 13072   ↾_s cress 13073  Rngcrng 13935  SubRngcsubrng 14201

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-ov 6016  df-inn 9134  df-ndx 13075  df-slot 13076  df-base 13078  df-subrng 14202

This theorem is referenced by:  subrngss  14204  subrngid  14205  subrngrng  14206  subrngrcl  14207  issubrng2  14214  subsubrng  14218  subrngpropd  14220  rng2idlsubrng  14521