Theorem aks6d1c1p1rcl 42539

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 1089	. . . . . 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 5153	. . . . 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 2760	. . . 4 ⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)))}
6		opabssxp 5714	. . . 4 ⊢ {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵) ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)))} ⊆ (ℕ × 𝐵)
7	5, 6	eqsstri 3969	. . 3 ⊢ ∼ ⊆ (ℕ × 𝐵)
8	7	brel 5687	. 2 ⊢ (𝐸 ∼ 𝐹 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵))
9	1, 8	syl 17	1 ⊢ (𝜑 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   class class class wbr 5086  {copab 5148   × cxp 5620  ‘cfv 6490  (class class class)co 7358  ℕcn 12146   PrimRoots cprimroots 42522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5628

This theorem is referenced by:  aks6d1c1p3  42541  aks6d1c1p4  42542  aks6d1c1p5  42543  aks6d1c1p6  42545  aks6d1c1p8  42546  aks6d1c2lem3  42557  aks6d1c2lem4  42558