Theorem subsubrg2 13461

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 13460	. . 3 |- ( A e. (SubRing ` R ) -> ( a e. (SubRing ` S ) <-> ( a e. (SubRing ` R ) /\ a C_ A ) ) )
3		elin 3330	. . . 4 |- ( a e. ( (SubRing ` R ) i^i ~P A ) <-> ( a e. (SubRing ` R ) /\ a e. ~P A ) )
4		velpw 3594	. . . . 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 2185	1 |- ( A e. (SubRing ` R ) -> (SubRing ` S ) = ( (SubRing ` R ) i^i ~P A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1363   e. wcel 2158   i^i cin 3140   C_ wss 3141   ~Pcpw 3587   ` cfv 5228  (class class class)co 5888   ↾_s cress 12476  SubRingcsubrg 13432

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-pre-ltirr 7936  ax-pre-lttrn 7938  ax-pre-ltadd 7940

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-ltxr 8010  df-inn 8933  df-2 8991  df-3 8992  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-iress 12483  df-plusg 12563  df-mulr 12564  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12839  df-subg 13061  df-mgp 13171  df-ur 13207  df-ring 13245  df-subrg 13434

This theorem is referenced by: (None)