Theorem subsubrg2 13560

Description: The set of subrings of a subring are the smaller subrings. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Hypothesis

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

Assertion

Ref	Expression
subsubrg2	|- ( A e. (SubRing ` R ) -> (SubRing ` S ) = ( (SubRing ` R ) i^i ~P A ) )

Proof of Theorem subsubrg2

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subsubrg.s	. . . 4 |- S = ( R ↾s A )
2	1	subsubrg 13559	. . 3 |- ( A e. (SubRing ` R ) -> ( a e. (SubRing ` S ) <-> ( a e. (SubRing ` R ) /\ a C_ A ) ) )
3		elin 3333	. . . 4 |- ( a e. ( (SubRing ` R ) i^i ~P A ) <-> ( a e. (SubRing ` R ) /\ a e. ~P A ) )
4		velpw 3597	. . . . 5 |- ( a e. ~P A <-> a C_ A )
5	4	anbi2i 457	. . . 4 |- ( ( a e. (SubRing ` R ) /\ a e. ~P A ) <-> ( a e. (SubRing ` R ) /\ a C_ A ) )
6	3, 5	bitr2i 185	. . 3 |- ( ( a e. (SubRing ` R ) /\ a C_ A ) <-> a e. ( (SubRing ` R ) i^i ~P A ) )
7	2, 6	bitrdi 196	. 2 |- ( A e. (SubRing ` R ) -> ( a e. (SubRing ` S ) <-> a e. ( (SubRing ` R ) i^i ~P A ) ) )
8	7	eqrdv 2187	1 |- ( A e. (SubRing ` R ) -> (SubRing ` S ) = ( (SubRing ` R ) i^i ~P A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   i^i cin 3143   C_ wss 3144   ~Pcpw 3590   ` cfv 5231  (class class class)co 5891   ↾_s cress 12487  SubRingcsubrg 13531

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-addcom 7930  ax-addass 7932  ax-i2m1 7935  ax-0lt1 7936  ax-0id 7938  ax-rnegex 7939  ax-pre-ltirr 7942  ax-pre-lttrn 7944  ax-pre-ltadd 7946

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-pnf 8013  df-mnf 8014  df-ltxr 8016  df-inn 8939  df-2 8997  df-3 8998  df-ndx 12489  df-slot 12490  df-base 12492  df-sets 12493  df-iress 12494  df-plusg 12574  df-mulr 12575  df-0g 12735  df-mgm 12804  df-sgrp 12837  df-mnd 12850  df-subg 13081  df-mgp 13242  df-ur 13281  df-ring 13319  df-subrg 13533

This theorem is referenced by: (None)