Theorem rrgmex 14073

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 4811	. . . 4 ⊢ Rel (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))})
2		df-rlreg 14070	. . . . 5 ⊢ RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))})
3	2	releqi 4763	. . . 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 2273	. . . 4 ⊢ (𝐴 ∈ 𝐸 ↔ 𝐴 ∈ (RLReg‘𝑅))
7	6	biimpi 120	. . 3 ⊢ (𝐴 ∈ 𝐸 → 𝐴 ∈ (RLReg‘𝑅))
8		relelfvdm 5618	. . 3 ⊢ ((Rel RLReg ∧ 𝐴 ∈ (RLReg‘𝑅)) → 𝑅 ∈ dom RLReg)
9	4, 7, 8	sylancr 414	. 2 ⊢ (𝐴 ∈ 𝐸 → 𝑅 ∈ dom RLReg)
10	9	elexd 2787	1 ⊢ (𝐴 ∈ 𝐸 → 𝑅 ∈ V)

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2177  ∀wral 2485  {crab 2489  Vcvv 2773   ↦ cmpt 4110  dom cdm 4680  Rel wrel 4685  ‘cfv 5277  (class class class)co 5954  Basecbs 12882  ._rcmulr 12960  0_gc0g 13138  RLRegcrlreg 14067

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-br 4049  df-opab 4111  df-mpt 4112  df-xp 4686  df-rel 4687  df-dm 4690  df-iota 5238  df-fv 5285  df-rlreg 14070

This theorem is referenced by:  rrgval  14074