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Theorem riinerm 6470
Description: The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
Assertion
Ref Expression
riinerm ((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 𝑅 Er 𝐵) → ((𝐵 × 𝐵) ∩ 𝑥𝐴 𝑅) Er 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑦,𝐴
Allowed substitution hints:   𝐵(𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem riinerm
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 iinerm 6469 . 2 ((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 𝑅 Er 𝐵) → 𝑥𝐴 𝑅 Er 𝐵)
2 eleq1 2180 . . . . . 6 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
32cbvexv 1872 . . . . 5 (∃𝑥 𝑥𝐴 ↔ ∃𝑎 𝑎𝐴)
4 eleq1 2180 . . . . . 6 (𝑎 = 𝑦 → (𝑎𝐴𝑦𝐴))
54cbvexv 1872 . . . . 5 (∃𝑎 𝑎𝐴 ↔ ∃𝑦 𝑦𝐴)
63, 5bitri 183 . . . 4 (∃𝑥 𝑥𝐴 ↔ ∃𝑦 𝑦𝐴)
7 erssxp 6420 . . . . . . 7 (𝑅 Er 𝐵𝑅 ⊆ (𝐵 × 𝐵))
87ralimi 2472 . . . . . 6 (∀𝑥𝐴 𝑅 Er 𝐵 → ∀𝑥𝐴 𝑅 ⊆ (𝐵 × 𝐵))
9 riinm 3855 . . . . . 6 ((∀𝑥𝐴 𝑅 ⊆ (𝐵 × 𝐵) ∧ ∃𝑥 𝑥𝐴) → ((𝐵 × 𝐵) ∩ 𝑥𝐴 𝑅) = 𝑥𝐴 𝑅)
108, 9sylan 281 . . . . 5 ((∀𝑥𝐴 𝑅 Er 𝐵 ∧ ∃𝑥 𝑥𝐴) → ((𝐵 × 𝐵) ∩ 𝑥𝐴 𝑅) = 𝑥𝐴 𝑅)
11 ereq1 6404 . . . . 5 (((𝐵 × 𝐵) ∩ 𝑥𝐴 𝑅) = 𝑥𝐴 𝑅 → (((𝐵 × 𝐵) ∩ 𝑥𝐴 𝑅) Er 𝐵 𝑥𝐴 𝑅 Er 𝐵))
1210, 11syl 14 . . . 4 ((∀𝑥𝐴 𝑅 Er 𝐵 ∧ ∃𝑥 𝑥𝐴) → (((𝐵 × 𝐵) ∩ 𝑥𝐴 𝑅) Er 𝐵 𝑥𝐴 𝑅 Er 𝐵))
136, 12sylan2br 286 . . 3 ((∀𝑥𝐴 𝑅 Er 𝐵 ∧ ∃𝑦 𝑦𝐴) → (((𝐵 × 𝐵) ∩ 𝑥𝐴 𝑅) Er 𝐵 𝑥𝐴 𝑅 Er 𝐵))
1413ancoms 266 . 2 ((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 𝑅 Er 𝐵) → (((𝐵 × 𝐵) ∩ 𝑥𝐴 𝑅) Er 𝐵 𝑥𝐴 𝑅 Er 𝐵))
151, 14mpbird 166 1 ((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 𝑅 Er 𝐵) → ((𝐵 × 𝐵) ∩ 𝑥𝐴 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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