Theorem brredunds 35895

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 3987	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵))
2	1	3adant3 1127	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵))
3		ineq12 4177	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝐶) → (𝑥 ∩ 𝑧) = (𝐴 ∩ 𝐶))
4	3	3adant2 1126	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥 ∩ 𝑧) = (𝐴 ∩ 𝐶))
5		ineq12 4177	. . . . 5 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦 ∩ 𝑧) = (𝐵 ∩ 𝐶))
6	5	3adant1 1125	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦 ∩ 𝑧) = (𝐵 ∩ 𝐶))
7	4, 6	eqeq12d 2836	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧) ↔ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)))
8	2, 7	anbi12d 632	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑥 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧)) ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))))
9		df-redunds 35892	. 2 ⊢ Redunds = ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ (𝑥 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧))}
10	8, 9	brcnvrabga 35633	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 Redunds ⟨𝐵, 𝐶⟩ ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1082   = wceq 1536   ∈ wcel 2113   ∩ cin 3928   ⊆ wss 3929  ⟨cop 4566   class class class wbr 5059   Redunds credunds 35507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pr 5323

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-br 5060  df-opab 5122  df-xp 5554  df-rel 5555  df-cnv 5556  df-oprab 7153  df-redunds 35892

This theorem is referenced by:  brredundsredund  35896