Theorem subrngrng 14206

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 1022	. 2 |- ( ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ ( Base ` R ) ) -> ( R ↾s A ) e. Rng )
2		eqid 2229	. . 3 |- ( Base ` R ) = ( Base ` R )
3	2	issubrng 14203	. 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 2295	. 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 1002   = wceq 1395   e. wcel 2200   C_ wss 3198   ` 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:  subrngsubg  14208  subrngmcl  14213  subsubrng  14218