Theorem relpropg 46111

Description: A pair of ordered pairs is a relation. (Contributed by Eric Schmidt, 24-Oct-2026.)

Assertion

Ref	Expression
relpropg	⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → Rel {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})

Proof of Theorem relpropg

Step	Hyp	Ref	Expression
1		relsnopg 5786	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Rel {⟨𝐴, 𝐵⟩})
2		relsnopg 5786	. . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → Rel {⟨𝐶, 𝐷⟩})
3	1, 2	anim12i 625	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → (Rel {⟨𝐴, 𝐵⟩} ∧ Rel {⟨𝐶, 𝐷⟩}))
4		relun 5794	. . 3 ⊢ (Rel ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) ↔ (Rel {⟨𝐴, 𝐵⟩} ∧ Rel {⟨𝐶, 𝐷⟩}))
5	3, 4	sylibr 237	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → Rel ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}))
6		df-pr 4591	. . 3 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
7	6	releqi 5759	. 2 ⊢ (Rel {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ↔ Rel ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}))
8	5, 7	sylibr 237	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → Rel {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147   ∪ cun 4152  {csn 4588  {cpr 4590  ⟨cop 4838  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-opab 5171  df-xp 5661  df-rel 5662

This theorem is used by:  grpcanonhfstruct  46228