Theorem aks6d1c1p1rcl 43159

Description: Reverse closure of the introspective relation. (Contributed by metakunt, 25-Apr-2025.)

Hypotheses

Ref	Expression
aks6d1c1p1rcl.1	⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)))}
aks6d1c1p1rcl.2	⊢ (𝜑 → 𝐸 ∼ 𝐹)

Assertion

Ref	Expression
aks6d1c1p1rcl	⊢ (𝜑 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵))

Distinct variable group:   𝐵,𝑒,𝑓

Allowed substitution hints:   𝜑(𝑦, 𝑒, 𝑓)   𝐵(𝑦)   𝐷(𝑦, 𝑒, 𝑓)   ∼ (𝑦, 𝑒, 𝑓)   𝑅(𝑦, 𝑒, 𝑓)   𝐸(𝑦, 𝑒, 𝑓)   ↑ (𝑦, 𝑒, 𝑓)   𝐹(𝑦, 𝑒, 𝑓)   𝐾(𝑦, 𝑒, 𝑓)   𝑂(𝑦, 𝑒, 𝑓)

Proof of Theorem aks6d1c1p1rcl

Step	Hyp	Ref	Expression
1		aks6d1c1p1rcl.2	. 2 ⊢ (𝜑 → 𝐸 ∼ 𝐹)
2		aks6d1c1p1rcl.1	. . . . 5 ⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)))}
3		df-3an 1105	. . . . . 6 ⊢ ((𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦))) ↔ ((𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦))))
4	3	opabbii 5171	. . . . 5 ⊢ {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)))} = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)))}
5	2, 4	eqtri 2783	. . . 4 ⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)))}
6		opabssxp 5742	. . . 4 ⊢ {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)))} ⊆ (ℕ × 𝐵)
7	5, 6	eqsstri 3954	. . 3 ⊢ ∼ ⊆ (ℕ × 𝐵)
8	7	brel 5715	. 2 ⊢ (𝐸 ∼ 𝐹 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵))
9	1, 8	syl 18	1 ⊢ (𝜑 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076   class class class wbr 5102  {copab 5166   × cxp 5648  ‘cfv 6538  (class class class)co 7420  ℕcn 12335   PrimRoots cprimroots 43142

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5656

This theorem is used by:  aks6d1c1p3  43161  aks6d1c1p4  43162  aks6d1c1p5  43163  aks6d1c1p6  43165  aks6d1c1p8  43166  aks6d1c2lem3  43177  aks6d1c2lem4  43178