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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  =  ( Rs  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.
StepHypRef Expression
1 subsubrg.s . . . 4  |-  S  =  ( Rs  A )
21subsubrg 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 )
54anbi2i 457 . . . 4  |-  ( ( a  e.  (SubRing `  R
)  /\  a  e.  ~P A )  <->  ( a  e.  (SubRing `  R )  /\  a  C_  A ) )
63, 5bitr2i 185 . . 3  |-  ( ( a  e.  (SubRing `  R
)  /\  a  C_  A )  <->  a  e.  ( (SubRing `  R )  i^i  ~P A ) )
72, 6bitrdi 196 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( a  e.  (SubRing `  S )  <->  a  e.  ( (SubRing `  R
)  i^i  ~P A
) ) )
87eqrdv 2187 1  |-  ( A  e.  (SubRing `  R
)  ->  (SubRing `  S
)  =  ( (SubRing `  R )  i^i  ~P A ) )
Colors of variables: wff set class
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)
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