Theorem subrngrng 14215

Description: A subring is a non-unital ring. (Contributed by AV, 14-Feb-2025.)

Hypothesis

Ref	Expression
subrngrng.1	|- S = ( R ↾s A )

Assertion

Ref	Expression
subrngrng	|- ( A e. (SubRng ` R ) -> S e. Rng )

Proof of Theorem subrngrng

Step	Hyp	Ref	Expression
1		simp2 1024	. 2 |- ( ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ ( Base ` R ) ) -> ( R ↾s A ) e. Rng )
2		eqid 2231	. . 3 |- ( Base ` R ) = ( Base ` R )
3	2	issubrng 14212	. 2 |- ( A e. (SubRng ` R ) <-> ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ ( Base ` R ) ) )
4		subrngrng.1	. . 3 |- S = ( R ↾s A )
5	4	eleq1i 2297	. 2 |- ( S e. Rng <-> ( R ↾s A ) e. Rng )
6	1, 3, 5	3imtr4i 201	1 |- ( A e. (SubRng ` R ) -> S e. Rng )

Syntax hints:   -> wi 4   /\ w3a 1004   = wceq 1397   e. wcel 2202   C_ wss 3200   ` cfv 5326  (class class class)co 6017   Basecbs 13081   ↾_s cress 13082  Rngcrng 13944  SubRngcsubrng 14210

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6020  df-inn 9143  df-ndx 13084  df-slot 13085  df-base 13087  df-subrng 14211

This theorem is referenced by:  subrngsubg  14217  subrngmcl  14222  subsubrng  14227