Theorem igenss 38063

Description: A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
igenval.1	⊢ 𝐺 = (1st ‘𝑅)
igenval.2	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
igenss	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆))

Proof of Theorem igenss

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssintub 4974	. 2 ⊢ 𝑆 ⊆ ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}
2		igenval.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
3		igenval.2	. . 3 ⊢ 𝑋 = ran 𝐺
4	2, 3	igenval 38062	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
5	1, 4	sseqtrrid 4052	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {crab 3436   ⊆ wss 3966  ∩ cint 4954  ran crn 5694  ‘cfv 6569  (class class class)co 7438  1^st c1st 8020  RingOpscrngo 37895  Idlcidl 38008   IdlGen cigen 38060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-int 4955  df-iun 5001  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-fo 6575  df-fv 6577  df-riota 7395  df-ov 7441  df-oprab 7442  df-mpo 7443  df-1st 8022  df-2nd 8023  df-grpo 30538  df-gid 30539  df-ablo 30590  df-rngo 37896  df-idl 38011  df-igen 38061

This theorem is referenced by:  igenval2  38067  isfldidl  38069  ispridlc  38071