Theorem issubrng 13961

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 13960	. . 3 |- SubRng = ( w e. Rng |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Rng } )
2	1	mptrcl 5662	. 2 |- ( A e. (SubRng ` R ) -> R e. Rng )
3		simp1 1000	. 2 |- ( ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) -> R e. Rng )
4		df-subrng 13960	. . . . 5 |- SubRng = ( r e. Rng |-> { s e. ~P ( Base ` r ) | ( r ↾s s ) e. Rng } )
5		fveq2 5576	. . . . . . 7 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
6	5	pweqd 3621	. . . . . 6 |- ( r = R -> ~P ( Base ` r ) = ~P ( Base ` R ) )
7		oveq1 5951	. . . . . . 7 |- ( r = R -> ( r ↾s s ) = ( R ↾s s ) )
8	7	eleq1d 2274	. . . . . 6 |- ( r = R -> ( ( r ↾s s ) e. Rng <-> ( R ↾s s ) e. Rng ) )
9	6, 8	rabeqbidv 2767	. . . . 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 12890	. . . . . . . 8 |- Base Fn _V
12		elex 2783	. . . . . . . 8 |- ( R e. Rng -> R e. _V )
13		funfvex 5593	. . . . . . . . 9 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
14	13	funfni 5376	. . . . . . . 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 4225	. . . . . 6 |- ( R e. Rng -> ~P ( Base ` R ) e. _V )
17		rabexg 4187	. . . . . 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 5673	. . . 4 |- ( R e. Rng -> (SubRng ` R ) = { s e. ~P ( Base ` R ) | ( R ↾s s ) e. Rng } )
20	19	eleq2d 2275	. . 3 |- ( R e. Rng -> ( A e. (SubRng ` R ) <-> A e. { s e. ~P ( Base ` R ) | ( R ↾s s ) e. Rng } ) )
21		oveq2 5952	. . . . . 6 |- ( s = A -> ( R ↾s s ) = ( R ↾s A ) )
22	21	eleq1d 2274	. . . . 5 |- ( s = A -> ( ( R ↾s s ) e. Rng <-> ( R ↾s A ) e. Rng ) )
23	22	elrab 2929	. . . 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 2209	. . . . . . . 8 |- ( Base ` R ) = B
26	25	sseq2i 3220	. . . . . . 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 4200	. . . . . . . 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 985	. . . . . 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 706	1 |- ( A e. (SubRng ` R ) <-> ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   {crab 2488   _Vcvv 2772   C_ wss 3166   ~Pcpw 3616   Fn wfn 5266   ` cfv 5271  (class class class)co 5944   Basecbs 12832   ↾_s cress 12833  Rngcrng 13694  SubRngcsubrng 13959

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5947  df-inn 9037  df-ndx 12835  df-slot 12836  df-base 12838  df-subrng 13960

This theorem is referenced by:  subrngss  13962  subrngid  13963  subrngrng  13964  subrngrcl  13965  issubrng2  13972  subsubrng  13976  subrngpropd  13978  rng2idlsubrng  14279