Theorem issubrng 14480

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 14479	. . 3 |- SubRng = ( w e. Rng |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Rng } )
2	1	mptrcl 5782	. 2 |- ( A e. (SubRng ` R ) -> R e. Rng )
3		simp1 1028	. 2 |- ( ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) -> R e. Rng )
4		df-subrng 14479	. . . . 5 |- SubRng = ( r e. Rng |-> { s e. ~P ( Base ` r ) | ( r ↾s s ) e. Rng } )
5		fveq2 5690	. . . . . . 7 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
6	5	pweqd 3690	. . . . . 6 |- ( r = R -> ~P ( Base ` r ) = ~P ( Base ` R ) )
7		oveq1 6082	. . . . . . 7 |- ( r = R -> ( r ↾s s ) = ( R ↾s s ) )
8	7	eleq1d 2307	. . . . . 6 |- ( r = R -> ( ( r ↾s s ) e. Rng <-> ( R ↾s s ) e. Rng ) )
9	6, 8	rabeqbidv 2816	. . . . 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 13389	. . . . . . . 8 |- Base Fn _V
12		elex 2833	. . . . . . . 8 |- ( R e. Rng -> R e. _V )
13		funfvex 5707	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
14	13	funfni 5478	. . . . . . . 8 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
15	11, 12, 14	sylancr 418	. . . . . . 7 |- ( R e. Rng -> ( Base ` R ) e. _V )
16	15	pwexd 4313	. . . . . 6 |- ( R e. Rng -> ~P ( Base ` R ) e. _V )
17		rabexg 4274	. . . . . 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 5793	. . . 4 |- ( R e. Rng -> (SubRng ` R ) = { s e. ~P ( Base ` R ) | ( R ↾s s ) e. Rng } )
20	19	eleq2d 2308	. . 3 |- ( R e. Rng -> ( A e. (SubRng ` R ) <-> A e. { s e. ~P ( Base ` R ) | ( R ↾s s ) e. Rng } ) )
21		oveq2 6083	. . . . . 6 |- ( s = A -> ( R ↾s s ) = ( R ↾s A ) )
22	21	eleq1d 2307	. . . . 5 |- ( s = A -> ( ( R ↾s s ) e. Rng <-> ( R ↾s A ) e. Rng ) )
23	22	elrab 2982	. . . 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 2242	. . . . . . . 8 |- ( Base ` R ) = B
26	25	sseq2i 3275	. . . . . . 7 |- ( A C_ ( Base ` R ) <-> A C_ B )
27	26	anbi2i 461	. . . . . 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 4287	. . . . . . . 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 468	. . . . . 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 1013	. . . . . 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 716	1 |- ( A e. (SubRng ` R ) <-> ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   C_ wss 3220   ~Pcpw 3685   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   Basecbs 13330   ↾_s cress 13331  Rngcrng 14206  SubRngcsubrng 14478

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-inn 9284  df-ndx 13333  df-slot 13334  df-base 13336  df-subrng 14479

This theorem is referenced by:  subrngss  14481  subrngid  14482  subrngrng  14483  subrngrcl  14484  issubrng2  14491  subsubrng  14495  subrngpropd  14497  rng2idlsubrng  14826