Theorem subsubrg2 13880

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 13879	. . 3 |- ( A e. (SubRing ` R ) -> ( a e. (SubRing ` S ) <-> ( a e. (SubRing ` R ) /\ a C_ A ) ) )
3		elin 3347	. . . 4 |- ( a e. ( (SubRing ` R ) i^i ~P A ) <-> ( a e. (SubRing ` R ) /\ a e. ~P A ) )
4		velpw 3613	. . . . 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 2194	1 |- ( A e. (SubRing ` R ) -> (SubRing ` S ) = ( (SubRing ` R ) i^i ~P A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   i^i cin 3156   C_ wss 3157   ~Pcpw 3606   ` cfv 5259  (class class class)co 5925   ↾_s cress 12706  SubRingcsubrg 13851

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-addcom 7998  ax-addass 8000  ax-i2m1 8003  ax-0lt1 8004  ax-0id 8006  ax-rnegex 8007  ax-pre-ltirr 8010  ax-pre-lttrn 8012  ax-pre-ltadd 8014

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8082  df-mnf 8083  df-ltxr 8085  df-inn 9010  df-2 9068  df-3 9069  df-ndx 12708  df-slot 12709  df-base 12711  df-sets 12712  df-iress 12713  df-plusg 12795  df-mulr 12796  df-0g 12962  df-mgm 13060  df-sgrp 13106  df-mnd 13121  df-subg 13378  df-mgp 13555  df-ur 13594  df-ring 13632  df-subrg 13853

This theorem is referenced by: (None)