Theorem rrgss 20563

Description: Left-regular elements are a subset of the base set. (Contributed by Stefan O'Rear, 22-Mar-2015.)

Hypotheses

Ref	Expression
rrgss.e	⊢ 𝐸 = (RLReg‘𝑅)
rrgss.b	⊢ 𝐵 = (Base‘𝑅)

Assertion

Ref	Expression
rrgss	⊢ 𝐸 ⊆ 𝐵

Proof of Theorem rrgss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrgss.e	. . 3 ⊢ 𝐸 = (RLReg‘𝑅)
2		rrgss.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
3		eqid 2738	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2738	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
5	1, 2, 3, 4	rrgval 20558	. 2 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → 𝑦 = (0g‘𝑅))}
6	5	ssrab3 4015	1 ⊢ 𝐸 ⊆ 𝐵

Syntax hints:   → wi 4   = wceq 1539  ∀wral 3064   ⊆ wss 3887  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  ._rcmulr 16963  0_gc0g 17150  RLRegcrlreg 20550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-rlreg 20554

This theorem is referenced by:  znrrg  20773  mdegvsca  25241  deg1mul3  25280