Theorem subsubrng2 14291

Description: The set of subrings of a subring are the smaller subrings. (Contributed by AV, 15-Feb-2025.)

Hypothesis

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

Assertion

Ref	Expression
subsubrng2	|- ( A e. (SubRng ` R ) -> (SubRng ` S ) = ( (SubRng ` R ) i^i ~P A ) )

Proof of Theorem subsubrng2

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subsubrng.s	. . . 4 |- S = ( R ↾s A )
2	1	subsubrng 14290	. . 3 |- ( A e. (SubRng ` R ) -> ( a e. (SubRng ` S ) <-> ( a e. (SubRng ` R ) /\ a C_ A ) ) )
3		elin 3392	. . . 4 |- ( a e. ( (SubRng ` R ) i^i ~P A ) <-> ( a e. (SubRng ` R ) /\ a e. ~P A ) )
4		velpw 3663	. . . . 5 |- ( a e. ~P A <-> a C_ A )
5	4	anbi2i 457	. . . 4 |- ( ( a e. (SubRng ` R ) /\ a e. ~P A ) <-> ( a e. (SubRng ` R ) /\ a C_ A ) )
6	3, 5	bitr2i 185	. . 3 |- ( ( a e. (SubRng ` R ) /\ a C_ A ) <-> a e. ( (SubRng ` R ) i^i ~P A ) )
7	2, 6	bitrdi 196	. 2 |- ( A e. (SubRng ` R ) -> ( a e. (SubRng ` S ) <-> a e. ( (SubRng ` R ) i^i ~P A ) ) )
8	7	eqrdv 2229	1 |- ( A e. (SubRng ` R ) -> (SubRng ` S ) = ( (SubRng ` R ) i^i ~P A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   i^i cin 3200   C_ wss 3201   ~Pcpw 3656   ` cfv 5333  (class class class)co 6028   ↾_s cress 13144  SubRngcsubrng 14273

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-inn 9187  df-2 9245  df-3 9246  df-ndx 13146  df-slot 13147  df-base 13149  df-sets 13150  df-iress 13151  df-plusg 13234  df-mulr 13235  df-subg 13818  df-abl 13935  df-rng 14008  df-subrng 14274

This theorem is referenced by: (None)