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Theorem subsubrg2 17010
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 17009 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  ( a  e.  (SubRing `  S )  <->  ( a  e.  (SubRing `  R
)  /\  a  C_  A ) ) )
3 elin 3642 . . . 4  |-  ( a  e.  ( (SubRing `  R
)  i^i  ~P A
)  <->  ( a  e.  (SubRing `  R )  /\  a  e.  ~P A ) )
4 selpw 3970 . . . . 5  |-  ( a  e.  ~P A  <->  a  C_  A )
54anbi2i 694 . . . 4  |-  ( ( a  e.  (SubRing `  R
)  /\  a  e.  ~P A )  <->  ( a  e.  (SubRing `  R )  /\  a  C_  A ) )
63, 5bitr2i 250 . . 3  |-  ( ( a  e.  (SubRing `  R
)  /\  a  C_  A )  <->  a  e.  ( (SubRing `  R )  i^i  ~P A ) )
72, 6syl6bb 261 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( a  e.  (SubRing `  S )  <->  a  e.  ( (SubRing `  R
)  i^i  ~P A
) ) )
87eqrdv 2449 1  |-  ( A  e.  (SubRing `  R
)  ->  (SubRing `  S
)  =  ( (SubRing `  R )  i^i  ~P A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    i^i cin 3430    C_ wss 3431   ~Pcpw 3963   ` cfv 5521  (class class class)co 6195   ↾s cress 14288  SubRingcsubrg 16979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-recs 6937  df-rdg 6971  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-3 10487  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-0g 14494  df-mnd 15529  df-subg 15792  df-mgp 16709  df-ur 16721  df-rng 16765  df-subrg 16981
This theorem is referenced by:  evlseu  17721
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