Theorem subrngrng 14049

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 1001	. 2 |- ( ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ ( Base ` R ) ) -> ( R ↾s A ) e. Rng )
2		eqid 2206	. . 3 |- ( Base ` R ) = ( Base ` R )
3	2	issubrng 14046	. 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 2272	. 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 981   = wceq 1373   e. wcel 2177   C_ wss 3170   ` cfv 5285  (class class class)co 5962   Basecbs 12917   ↾_s cress 12918  Rngcrng 13779  SubRngcsubrng 14044

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-cnex 8046  ax-resscn 8047  ax-1re 8049  ax-addrcl 8052

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-fv 5293  df-ov 5965  df-inn 9067  df-ndx 12920  df-slot 12921  df-base 12923  df-subrng 14045

This theorem is referenced by:  subrngsubg  14051  subrngmcl  14056  subsubrng  14061