Theorem rrgmex 14492

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 4888	. . . 4 ⊢ Rel (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))})
2		df-rlreg 14489	. . . . 5 ⊢ RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))})
3	2	releqi 4838	. . . 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 2301	. . . 4 ⊢ (𝐴 ∈ 𝐸 ↔ 𝐴 ∈ (RLReg‘𝑅))
7	6	biimpi 120	. . 3 ⊢ (𝐴 ∈ 𝐸 → 𝐴 ∈ (RLReg‘𝑅))
8		relelfvdm 5707	. . 3 ⊢ ((Rel RLReg ∧ 𝐴 ∈ (RLReg‘𝑅)) → 𝑅 ∈ dom RLReg)
9	4, 7, 8	sylancr 414	. 2 ⊢ (𝐴 ∈ 𝐸 → 𝑅 ∈ dom RLReg)
10	9	elexd 2829	1 ⊢ (𝐴 ∈ 𝐸 → 𝑅 ∈ V)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2205  ∀wral 2522  {crab 2526  Vcvv 2815   ↦ cmpt 4176  dom cdm 4754  Rel wrel 4759  ‘cfv 5357  (class class class)co 6058  Basecbs 13296  ._rcmulr 13375  0_gc0g 13553  RLRegcrlreg 14486

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-xp 4760  df-rel 4761  df-dm 4764  df-iota 5317  df-fv 5365  df-rlreg 14489

This theorem is referenced by:  rrgval  14493