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Mirrors > Home > ILE Home > Th. List > riinerm | GIF version |
Description: The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.) |
Ref | Expression |
---|---|
riinerm | ⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑅 Er 𝐵) → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iinerm 6469 | . 2 ⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑅 Er 𝐵) → ∩ 𝑥 ∈ 𝐴 𝑅 Er 𝐵) | |
2 | eleq1 2180 | . . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴)) | |
3 | 2 | cbvexv 1872 | . . . . 5 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑎 𝑎 ∈ 𝐴) |
4 | eleq1 2180 | . . . . . 6 ⊢ (𝑎 = 𝑦 → (𝑎 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
5 | 4 | cbvexv 1872 | . . . . 5 ⊢ (∃𝑎 𝑎 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴) |
6 | 3, 5 | bitri 183 | . . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴) |
7 | erssxp 6420 | . . . . . . 7 ⊢ (𝑅 Er 𝐵 → 𝑅 ⊆ (𝐵 × 𝐵)) | |
8 | 7 | ralimi 2472 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 → ∀𝑥 ∈ 𝐴 𝑅 ⊆ (𝐵 × 𝐵)) |
9 | riinm 3855 | . . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝑅 ⊆ (𝐵 × 𝐵) ∧ ∃𝑥 𝑥 ∈ 𝐴) → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) = ∩ 𝑥 ∈ 𝐴 𝑅) | |
10 | 8, 9 | sylan 281 | . . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) = ∩ 𝑥 ∈ 𝐴 𝑅) |
11 | ereq1 6404 | . . . . 5 ⊢ (((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) = ∩ 𝑥 ∈ 𝐴 𝑅 → (((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵 ↔ ∩ 𝑥 ∈ 𝐴 𝑅 Er 𝐵)) | |
12 | 10, 11 | syl 14 | . . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴) → (((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵 ↔ ∩ 𝑥 ∈ 𝐴 𝑅 Er 𝐵)) |
13 | 6, 12 | sylan2br 286 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 ∧ ∃𝑦 𝑦 ∈ 𝐴) → (((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵 ↔ ∩ 𝑥 ∈ 𝐴 𝑅 Er 𝐵)) |
14 | 13 | ancoms 266 | . 2 ⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑅 Er 𝐵) → (((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵 ↔ ∩ 𝑥 ∈ 𝐴 𝑅 Er 𝐵)) |
15 | 1, 14 | mpbird 166 | 1 ⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑅 Er 𝐵) → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1316 ∃wex 1453 ∈ wcel 1465 ∀wral 2393 ∩ cin 3040 ⊆ wss 3041 ∩ ciin 3784 × cxp 4507 Er wer 6394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-iin 3786 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-er 6397 |
This theorem is referenced by: (None) |
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