Theorem rrgmex 13757

Description: A structure whose set of left-regular elements is inhabited is a set. (Contributed by Jim Kingdon, 12-Aug-2025.)

Hypothesis

Ref	Expression
rrgmex.e	⊢ 𝐸 = (RLReg‘𝑅)

Assertion

Ref	Expression
rrgmex	⊢ (𝐴 ∈ 𝐸 → 𝑅 ∈ V)

Proof of Theorem rrgmex

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

Step	Hyp	Ref	Expression
1		mptrel 4790	. . . 4 ⊢ Rel (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))})
2		df-rlreg 13754	. . . . 5 ⊢ RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))})
3	2	releqi 4742	. . . 4 ⊢ (Rel RLReg ↔ Rel (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))}))
4	1, 3	mpbir 146	. . 3 ⊢ Rel RLReg
5		rrgmex.e	. . . . 5 ⊢ 𝐸 = (RLReg‘𝑅)
6	5	eleq2i 2260	. . . 4 ⊢ (𝐴 ∈ 𝐸 ↔ 𝐴 ∈ (RLReg‘𝑅))
7	6	biimpi 120	. . 3 ⊢ (𝐴 ∈ 𝐸 → 𝐴 ∈ (RLReg‘𝑅))
8		relelfvdm 5586	. . 3 ⊢ ((Rel RLReg ∧ 𝐴 ∈ (RLReg‘𝑅)) → 𝑅 ∈ dom RLReg)
9	4, 7, 8	sylancr 414	. 2 ⊢ (𝐴 ∈ 𝐸 → 𝑅 ∈ dom RLReg)
10	9	elexd 2773	1 ⊢ (𝐴 ∈ 𝐸 → 𝑅 ∈ V)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164  ∀wral 2472  {crab 2476  Vcvv 2760   ↦ cmpt 4090  dom cdm 4659  Rel wrel 4664  ‘cfv 5254  (class class class)co 5918  Basecbs 12618  ._rcmulr 12696  0_gc0g 12867  RLRegcrlreg 13751

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-xp 4665  df-rel 4666  df-dm 4669  df-iota 5215  df-fv 5262  df-rlreg 13754

This theorem is referenced by:  rrgval  13758