Theorem brredunds 39445

Description: Binary relation on the class of all redundant sets. (Contributed by Peter Mazsa, 25-Oct-2022.)

Assertion

Ref	Expression
brredunds	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 Redunds ⟨𝐵, 𝐶⟩ ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))))

Proof of Theorem brredunds

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq12 3961	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵))
2	1	3adant3 1150	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵))
3		ineq12 4164	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝐶) → (𝑥 ∩ 𝑧) = (𝐴 ∩ 𝐶))
4	3	3adant2 1149	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥 ∩ 𝑧) = (𝐴 ∩ 𝐶))
5		ineq12 4164	. . . . 5 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦 ∩ 𝑧) = (𝐵 ∩ 𝐶))
6	5	3adant1 1148	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦 ∩ 𝑧) = (𝐵 ∩ 𝐶))
7	4, 6	eqeq12d 2778	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧) ↔ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)))
8	2, 7	anbi12d 644	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑥 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧)) ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))))
9		df-redunds 39442	. 2 ⊢ Redunds = ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ (𝑥 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧))}
10	8, 9	brcnvrabga 39077	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 Redunds ⟨𝐵, 𝐶⟩ ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ∩ cin 3901   ⊆ wss 3902  ⟨cop 4593   class class class wbr 5107   Redunds credunds 38938

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

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 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-oprab 7420  df-redunds 39442

This theorem is used by:  brredundsredund  39446