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Theorem riinerm 6568
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 6567 . 2 ((∃𝑦 𝑦𝐴 ∧ ∀𝑥𝐴 𝑅 Er 𝐵) → 𝑥𝐴 𝑅 Er 𝐵)
2 eleq1 2227 . . . . . 6 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
32cbvexv 1905 . . . . 5 (∃𝑥 𝑥𝐴 ↔ ∃𝑎 𝑎𝐴)
4 eleq1 2227 . . . . . 6 (𝑎 = 𝑦 → (𝑎𝐴𝑦𝐴))
54cbvexv 1905 . . . . 5 (∃𝑎 𝑎𝐴 ↔ ∃𝑦 𝑦𝐴)
63, 5bitri 183 . . . 4 (∃𝑥 𝑥𝐴 ↔ ∃𝑦 𝑦𝐴)
7 erssxp 6518 . . . . . . 7 (𝑅 Er 𝐵𝑅 ⊆ (𝐵 × 𝐵))
87ralimi 2527 . . . . . 6 (∀𝑥𝐴 𝑅 Er 𝐵 → ∀𝑥𝐴 𝑅 ⊆ (𝐵 × 𝐵))
9 riinm 3935 . . . . . 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 6502 . . . . 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 1342  wex 1479  wcel 2135  wral 2442  cin 3113  wss 3114   ciin 3864   × cxp 4599   Er wer 6492
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-pow 4150  ax-pr 4184
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2726  df-un 3118  df-in 3120  df-ss 3127  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-iin 3866  df-br 3980  df-opab 4041  df-xp 4607  df-rel 4608  df-cnv 4609  df-co 4610  df-dm 4611  df-rn 4612  df-er 6495
This theorem is referenced by: (None)
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