Theorem subsubrng2 14027

Description: The set of subrings of a subring are the smaller subrings. (Contributed by AV, 15-Feb-2025.)

Hypothesis

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

Assertion

Ref	Expression
subsubrng2	|- ( A e. (SubRng ` R ) -> (SubRng ` S ) = ( (SubRng ` R ) i^i ~P A ) )

Proof of Theorem subsubrng2

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subsubrng.s	. . . 4 |- S = ( R ↾s A )
2	1	subsubrng 14026	. . 3 |- ( A e. (SubRng ` R ) -> ( a e. (SubRng ` S ) <-> ( a e. (SubRng ` R ) /\ a C_ A ) ) )
3		elin 3358	. . . 4 |- ( a e. ( (SubRng ` R ) i^i ~P A ) <-> ( a e. (SubRng ` R ) /\ a e. ~P A ) )
4		velpw 3625	. . . . 5 |- ( a e. ~P A <-> a C_ A )
5	4	anbi2i 457	. . . 4 |- ( ( a e. (SubRng ` R ) /\ a e. ~P A ) <-> ( a e. (SubRng ` R ) /\ a C_ A ) )
6	3, 5	bitr2i 185	. . 3 |- ( ( a e. (SubRng ` R ) /\ a C_ A ) <-> a e. ( (SubRng ` R ) i^i ~P A ) )
7	2, 6	bitrdi 196	. 2 |- ( A e. (SubRng ` R ) -> ( a e. (SubRng ` S ) <-> a e. ( (SubRng ` R ) i^i ~P A ) ) )
8	7	eqrdv 2204	1 |- ( A e. (SubRng ` R ) -> (SubRng ` S ) = ( (SubRng ` R ) i^i ~P A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   i^i cin 3167   C_ wss 3168   ~Pcpw 3618   ` cfv 5277  (class class class)co 5954   ↾_s cress 12883  SubRngcsubrng 14009

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-in1 615  ax-in2 616  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 4167  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-cnex 8029  ax-resscn 8030  ax-1re 8032  ax-addrcl 8035

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  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-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-br 4049  df-opab 4111  df-mpt 4112  df-id 4345  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-fv 5285  df-ov 5957  df-oprab 5958  df-mpo 5959  df-inn 9050  df-2 9108  df-3 9109  df-ndx 12885  df-slot 12886  df-base 12888  df-sets 12889  df-iress 12890  df-plusg 12972  df-mulr 12973  df-subg 13556  df-abl 13673  df-rng 13745  df-subrng 14010

This theorem is referenced by: (None)