Theorem issubrng 14445

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 14444	. . 3 |- SubRng = ( w e. Rng |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Rng } )
2	1	mptrcl 5765	. 2 |- ( A e. (SubRng ` R ) -> R e. Rng )
3		simp1 1024	. 2 |- ( ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) -> R e. Rng )
4		df-subrng 14444	. . . . 5 |- SubRng = ( r e. Rng |-> { s e. ~P ( Base ` r ) | ( r ↾s s ) e. Rng } )
5		fveq2 5675	. . . . . . 7 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
6	5	pweqd 3679	. . . . . 6 |- ( r = R -> ~P ( Base ` r ) = ~P ( Base ` R ) )
7		oveq1 6065	. . . . . . 7 |- ( r = R -> ( r ↾s s ) = ( R ↾s s ) )
8	7	eleq1d 2303	. . . . . 6 |- ( r = R -> ( ( r ↾s s ) e. Rng <-> ( R ↾s s ) e. Rng ) )
9	6, 8	rabeqbidv 2810	. . . . 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 13355	. . . . . . . 8 |- Base Fn _V
12		elex 2827	. . . . . . . 8 |- ( R e. Rng -> R e. _V )
13		funfvex 5692	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
14	13	funfni 5463	. . . . . . . 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 4299	. . . . . 6 |- ( R e. Rng -> ~P ( Base ` R ) e. _V )
17		rabexg 4260	. . . . . 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 5776	. . . 4 |- ( R e. Rng -> (SubRng ` R ) = { s e. ~P ( Base ` R ) | ( R ↾s s ) e. Rng } )
20	19	eleq2d 2304	. . 3 |- ( R e. Rng -> ( A e. (SubRng ` R ) <-> A e. { s e. ~P ( Base ` R ) | ( R ↾s s ) e. Rng } ) )
21		oveq2 6066	. . . . . 6 |- ( s = A -> ( R ↾s s ) = ( R ↾s A ) )
22	21	eleq1d 2303	. . . . 5 |- ( s = A -> ( ( R ↾s s ) e. Rng <-> ( R ↾s A ) e. Rng ) )
23	22	elrab 2976	. . . 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 2238	. . . . . . . 8 |- ( Base ` R ) = B
26	25	sseq2i 3269	. . . . . . 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 4273	. . . . . . . 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 1009	. . . . . 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 712	1 |- ( A e. (SubRng ` R ) <-> ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   {crab 2526   _Vcvv 2815   C_ wss 3214   ~Pcpw 3674   Fn wfn 5352   ` cfv 5357  (class class class)co 6058   Basecbs 13296   ↾_s cress 13297  Rngcrng 14171  SubRngcsubrng 14443

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-ov 6061  df-inn 9255  df-ndx 13299  df-slot 13300  df-base 13302  df-subrng 14444

This theorem is referenced by:  subrngss  14446  subrngid  14447  subrngrng  14448  subrngrcl  14449  issubrng2  14456  subsubrng  14460  subrngpropd  14462  rng2idlsubrng  14791