Theorem subsubrng2 14466

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 14465	. . 3 |- ( A e. (SubRng ` R ) -> ( a e. (SubRng ` S ) <-> ( a e. (SubRng ` R ) /\ a C_ A ) ) )
3		elin 3406	. . . 4 |- ( a e. ( (SubRng ` R ) i^i ~P A ) <-> ( a e. (SubRng ` R ) /\ a e. ~P A ) )
4		velpw 3682	. . . . 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 2232	1 |- ( A e. (SubRng ` R ) -> (SubRng ` S ) = ( (SubRng ` R ) i^i ~P A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   i^i cin 3213   C_ wss 3214   ~Pcpw 3675   ` cfv 5358  (class class class)co 6059   ↾_s cress 13302  SubRngcsubrng 14448

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-cnex 8235  ax-resscn 8236  ax-1re 8238  ax-addrcl 8241

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-br 4116  df-opab 4178  df-mpt 4179  df-id 4420  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-fv 5366  df-ov 6062  df-oprab 6063  df-mpo 6064  df-inn 9259  df-2 9317  df-3 9318  df-ndx 13304  df-slot 13305  df-base 13307  df-sets 13308  df-iress 13309  df-plusg 13392  df-mulr 13393  df-subg 13928  df-abl 14045  df-rng 14177  df-subrng 14449

This theorem is referenced by: (None)