Theorem rrgss 21110

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 2730	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2730	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
5	1, 2, 3, 4	rrgval 21105	. 2 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → 𝑦 = (0g‘𝑅))}
6	5	ssrab3 4081	1 ⊢ 𝐸 ⊆ 𝐵

Syntax hints:   → wi 4   = wceq 1539  ∀wral 3059   ⊆ wss 3949  ‘cfv 6544  (class class class)co 7413  Basecbs 17150  ._rcmulr 17204  0_gc0g 17391  RLRegcrlreg 21097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7416  df-rlreg 21101

This theorem is referenced by:  znrrg  21342  mdegvsca  25828  deg1mul3  25867