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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 6114 | . 2 ⊢ ((∃y y ∈ A ∧ ∀x ∈ A 𝑅 Er B) → ∩ x ∈ A 𝑅 Er B) | |
2 | eleq1 2097 | . . . . . 6 ⊢ (x = 𝑎 → (x ∈ A ↔ 𝑎 ∈ A)) | |
3 | 2 | cbvexv 1792 | . . . . 5 ⊢ (∃x x ∈ A ↔ ∃𝑎 𝑎 ∈ A) |
4 | eleq1 2097 | . . . . . 6 ⊢ (𝑎 = y → (𝑎 ∈ A ↔ y ∈ A)) | |
5 | 4 | cbvexv 1792 | . . . . 5 ⊢ (∃𝑎 𝑎 ∈ A ↔ ∃y y ∈ A) |
6 | 3, 5 | bitri 173 | . . . 4 ⊢ (∃x x ∈ A ↔ ∃y y ∈ A) |
7 | erssxp 6065 | . . . . . . 7 ⊢ (𝑅 Er B → 𝑅 ⊆ (B × B)) | |
8 | 7 | ralimi 2378 | . . . . . 6 ⊢ (∀x ∈ A 𝑅 Er B → ∀x ∈ A 𝑅 ⊆ (B × B)) |
9 | riinm 3720 | . . . . . 6 ⊢ ((∀x ∈ A 𝑅 ⊆ (B × B) ∧ ∃x x ∈ A) → ((B × B) ∩ ∩ x ∈ A 𝑅) = ∩ x ∈ A 𝑅) | |
10 | 8, 9 | sylan 267 | . . . . 5 ⊢ ((∀x ∈ A 𝑅 Er B ∧ ∃x x ∈ A) → ((B × B) ∩ ∩ x ∈ A 𝑅) = ∩ x ∈ A 𝑅) |
11 | ereq1 6049 | . . . . 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 272 | . . 3 ⊢ ((∀x ∈ A 𝑅 Er B ∧ ∃y y ∈ A) → (((B × B) ∩ ∩ x ∈ A 𝑅) Er B ↔ ∩ x ∈ A 𝑅 Er B)) |
14 | 13 | ancoms 255 | . 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 156 | 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 97 ↔ wb 98 = wceq 1242 ∃wex 1378 ∈ wcel 1390 ∀wral 2300 ∩ cin 2910 ⊆ wss 2911 ∩ ciin 3649 × cxp 4286 Er wer 6039 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-iin 3651 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-er 6042 |
This theorem is referenced by: (None) |
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