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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 6294 | . 2 ⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑅 Er 𝐵) → ∩ 𝑥 ∈ 𝐴 𝑅 Er 𝐵) | |
2 | eleq1 2145 | . . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴)) | |
3 | 2 | cbvexv 1838 | . . . . 5 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑎 𝑎 ∈ 𝐴) |
4 | eleq1 2145 | . . . . . 6 ⊢ (𝑎 = 𝑦 → (𝑎 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
5 | 4 | cbvexv 1838 | . . . . 5 ⊢ (∃𝑎 𝑎 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴) |
6 | 3, 5 | bitri 182 | . . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴) |
7 | erssxp 6245 | . . . . . . 7 ⊢ (𝑅 Er 𝐵 → 𝑅 ⊆ (𝐵 × 𝐵)) | |
8 | 7 | ralimi 2432 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 → ∀𝑥 ∈ 𝐴 𝑅 ⊆ (𝐵 × 𝐵)) |
9 | riinm 3776 | . . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝑅 ⊆ (𝐵 × 𝐵) ∧ ∃𝑥 𝑥 ∈ 𝐴) → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) = ∩ 𝑥 ∈ 𝐴 𝑅) | |
10 | 8, 9 | sylan 277 | . . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) = ∩ 𝑥 ∈ 𝐴 𝑅) |
11 | ereq1 6229 | . . . . 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 282 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 ∧ ∃𝑦 𝑦 ∈ 𝐴) → (((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵 ↔ ∩ 𝑥 ∈ 𝐴 𝑅 Er 𝐵)) |
14 | 13 | ancoms 264 | . 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 165 | 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 102 ↔ wb 103 = wceq 1285 ∃wex 1422 ∈ wcel 1434 ∀wral 2353 ∩ cin 2983 ⊆ wss 2984 ∩ ciin 3705 × cxp 4399 Er wer 6219 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-iin 3707 df-br 3812 df-opab 3866 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-er 6222 |
This theorem is referenced by: (None) |
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