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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 6625 | . 2 ⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑅 Er 𝐵) → ∩ 𝑥 ∈ 𝐴 𝑅 Er 𝐵) | |
2 | eleq1 2252 | . . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴)) | |
3 | 2 | cbvexv 1930 | . . . . 5 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑎 𝑎 ∈ 𝐴) |
4 | eleq1 2252 | . . . . . 6 ⊢ (𝑎 = 𝑦 → (𝑎 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
5 | 4 | cbvexv 1930 | . . . . 5 ⊢ (∃𝑎 𝑎 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴) |
6 | 3, 5 | bitri 184 | . . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴) |
7 | erssxp 6576 | . . . . . . 7 ⊢ (𝑅 Er 𝐵 → 𝑅 ⊆ (𝐵 × 𝐵)) | |
8 | 7 | ralimi 2553 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 → ∀𝑥 ∈ 𝐴 𝑅 ⊆ (𝐵 × 𝐵)) |
9 | riinm 3974 | . . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝑅 ⊆ (𝐵 × 𝐵) ∧ ∃𝑥 𝑥 ∈ 𝐴) → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) = ∩ 𝑥 ∈ 𝐴 𝑅) | |
10 | 8, 9 | sylan 283 | . . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) = ∩ 𝑥 ∈ 𝐴 𝑅) |
11 | ereq1 6560 | . . . . 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 288 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝑅 Er 𝐵 ∧ ∃𝑦 𝑦 ∈ 𝐴) → (((𝐵 × 𝐵) ∩ ∩ 𝑥 ∈ 𝐴 𝑅) Er 𝐵 ↔ ∩ 𝑥 ∈ 𝐴 𝑅 Er 𝐵)) |
14 | 13 | ancoms 268 | . 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 167 | 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 104 ↔ wb 105 = wceq 1364 ∃wex 1503 ∈ wcel 2160 ∀wral 2468 ∩ cin 3143 ⊆ wss 3144 ∩ ciin 3902 × cxp 4639 Er wer 6550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-iin 3904 df-br 4019 df-opab 4080 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-er 6553 |
This theorem is referenced by: (None) |
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