Theorem brredunds 38009

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 4004	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵))
2	1	3adant3 1129	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵))
3		ineq12 4202	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝐶) → (𝑥 ∩ 𝑧) = (𝐴 ∩ 𝐶))
4	3	3adant2 1128	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥 ∩ 𝑧) = (𝐴 ∩ 𝐶))
5		ineq12 4202	. . . . 5 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦 ∩ 𝑧) = (𝐵 ∩ 𝐶))
6	5	3adant1 1127	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦 ∩ 𝑧) = (𝐵 ∩ 𝐶))
7	4, 6	eqeq12d 2742	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧) ↔ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)))
8	2, 7	anbi12d 630	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑥 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧)) ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))))
9		df-redunds 38006	. 2 ⊢ Redunds = ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ (𝑥 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧))}
10	8, 9	brcnvrabga 37724	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 Redunds ⟨𝐵, 𝐶⟩ ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ∩ cin 3942   ⊆ wss 3943  ⟨cop 4629   class class class wbr 5141   Redunds credunds 37576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-oprab 7409  df-redunds 38006

This theorem is referenced by:  brredundsredund  38010