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| Description: The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.) | 
| Ref | Expression | 
|---|---|
| riiner | ⊢ (∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | xpider 8828 | . . 3 ⊢ (𝐵 × 𝐵) Er 𝐵 | |
| 2 | riin0 5082 | . . . . 5 ⊢ (𝐴 = ∅ → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) = (𝐵 × 𝐵)) | |
| 3 | 2 | adantl 481 | . . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 ∧ 𝐴 = ∅) → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) = (𝐵 × 𝐵)) | 
| 4 | ereq1 8752 | . . . 4 ⊢ (((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) = (𝐵 × 𝐵) → (((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵 ↔ (𝐵 × 𝐵) Er 𝐵)) | |
| 5 | 3, 4 | syl 17 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 ∧ 𝐴 = ∅) → (((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵 ↔ (𝐵 × 𝐵) Er 𝐵)) | 
| 6 | 1, 5 | mpbiri 258 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 ∧ 𝐴 = ∅) → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵) | 
| 7 | iiner 8829 | . . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝑅 Er 𝐵) → ∩ 𝑥 ∈ 𝐴 𝑅 Er 𝐵) | |
| 8 | 7 | ancoms 458 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 ∧ 𝐴 ≠ ∅) → ∩ 𝑥 ∈ 𝐴 𝑅 Er 𝐵) | 
| 9 | erssxp 8768 | . . . . . 6 ⊢ (𝑅 Er 𝐵 → 𝑅 ⊆ (𝐵 × 𝐵)) | |
| 10 | 9 | ralimi 3083 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 → ∀𝑥 ∈ 𝐴 𝑅 ⊆ (𝐵 × 𝐵)) | 
| 11 | riinn0 5083 | . . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝑅 ⊆ (𝐵 × 𝐵) ∧ 𝐴 ≠ ∅) → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) = ∩ 𝑥 ∈ 𝐴 𝑅) | |
| 12 | 10, 11 | sylan 580 | . . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 ∧ 𝐴 ≠ ∅) → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) = ∩ 𝑥 ∈ 𝐴 𝑅) | 
| 13 | ereq1 8752 | . . . 4 ⊢ (((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) = ∩ 𝑥 ∈ 𝐴 𝑅 → (((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵 ↔ ∩ 𝑥 ∈ 𝐴 𝑅 Er 𝐵)) | |
| 14 | 12, 13 | syl 17 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 ∧ 𝐴 ≠ ∅) → (((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵 ↔ ∩ 𝑥 ∈ 𝐴 𝑅 Er 𝐵)) | 
| 15 | 8, 14 | mpbird 257 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 ∧ 𝐴 ≠ ∅) → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵) | 
| 16 | 6, 15 | pm2.61dane 3029 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ≠ wne 2940 ∀wral 3061 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 ∩ ciin 4992 × cxp 5683 Er wer 8742 | 
| 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-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-ne 2941 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-iin 4994 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-er 8745 | 
| This theorem is referenced by: (None) | 
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