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Theorem riinerm 6115
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 y A x A 𝑅 Er B) → ((B × B) ∩ x A 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 6114 . 2 ((y y A x A 𝑅 Er B) → x A 𝑅 Er B)
2 eleq1 2097 . . . . . 6 (x = 𝑎 → (x A𝑎 A))
32cbvexv 1792 . . . . 5 (x x A𝑎 𝑎 A)
4 eleq1 2097 . . . . . 6 (𝑎 = y → (𝑎 Ay A))
54cbvexv 1792 . . . . 5 (𝑎 𝑎 Ay y A)
63, 5bitri 173 . . . 4 (x x Ay y A)
7 erssxp 6065 . . . . . . 7 (𝑅 Er B𝑅 ⊆ (B × B))
87ralimi 2378 . . . . . 6 (x A 𝑅 Er Bx A 𝑅 ⊆ (B × B))
9 riinm 3720 . . . . . 6 ((x A 𝑅 ⊆ (B × B) x x A) → ((B × B) ∩ x A 𝑅) = x A 𝑅)
108, 9sylan 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))
1210, 11syl 14 . . . 4 ((x A 𝑅 Er B x x A) → (((B × B) ∩ x A 𝑅) Er B x A 𝑅 Er B))
136, 12sylan2br 272 . . 3 ((x A 𝑅 Er B y y A) → (((B × B) ∩ x A 𝑅) Er B x A 𝑅 Er B))
1413ancoms 255 . 2 ((y y A x A 𝑅 Er B) → (((B × B) ∩ x A 𝑅) Er B x A 𝑅 Er B))
151, 14mpbird 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-bnd 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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