Theorem igenss 38265

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 4921	. 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 38264	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗})
5	1, 4	sseqtrrid 3977	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3399   ⊆ wss 3901  ∩ cint 4902  ran crn 5625  ‘cfv 6492  (class class class)co 7358  1^st c1st 7931  RingOpscrngo 38097  Idlcidl 38210   IdlGen cigen 38262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-grpo 30570  df-gid 30571  df-ablo 30622  df-rngo 38098  df-idl 38213  df-igen 38263

This theorem is referenced by:  igenval2  38269  isfldidl  38271  ispridlc  38273