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| Mirrors > Home > MPE Home > Th. List > Mathboxes > brredunds | Structured version Visualization version GIF version | ||
| Description: Binary relation on the class of all redundant sets. (Contributed by Peter Mazsa, 25-Oct-2022.) | 
| Ref | Expression | 
|---|---|
| brredunds | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 Redunds 〈𝐵, 𝐶〉 ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sseq12 4011 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵)) | |
| 2 | 1 | 3adant3 1133 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵)) | 
| 3 | ineq12 4215 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝐶) → (𝑥 ∩ 𝑧) = (𝐴 ∩ 𝐶)) | |
| 4 | 3 | 3adant2 1132 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥 ∩ 𝑧) = (𝐴 ∩ 𝐶)) | 
| 5 | ineq12 4215 | . . . . 5 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦 ∩ 𝑧) = (𝐵 ∩ 𝐶)) | |
| 6 | 5 | 3adant1 1131 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦 ∩ 𝑧) = (𝐵 ∩ 𝐶)) | 
| 7 | 4, 6 | eqeq12d 2753 | . . 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 38624 | . 2 ⊢ Redunds = ◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ (𝑥 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧))} | |
| 10 | 8, 9 | brcnvrabga 38343 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 Redunds 〈𝐵, 𝐶〉 ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 〈cop 4632 class class class wbr 5143 Redunds credunds 38202 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-oprab 7435 df-redunds 38624 | 
| This theorem is referenced by: brredundsredund 38628 | 
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