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Theorem erinxp 6635
Description: A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
erinxp.r (𝜑𝑅 Er 𝐴)
erinxp.a (𝜑𝐵𝐴)
Assertion
Ref Expression
erinxp (𝜑 → (𝑅 ∩ (𝐵 × 𝐵)) Er 𝐵)

Proof of Theorem erinxp
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss2 3371 . . . 4 (𝑅 ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
2 relxp 4753 . . . 4 Rel (𝐵 × 𝐵)
3 relss 4731 . . . 4 ((𝑅 ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵) → (Rel (𝐵 × 𝐵) → Rel (𝑅 ∩ (𝐵 × 𝐵))))
41, 2, 3mp2 16 . . 3 Rel (𝑅 ∩ (𝐵 × 𝐵))
54a1i 9 . 2 (𝜑 → Rel (𝑅 ∩ (𝐵 × 𝐵)))
6 simpr 110 . . . . 5 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦)
7 brinxp2 4711 . . . . 5 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ↔ (𝑥𝐵𝑦𝐵𝑥𝑅𝑦))
86, 7sylib 122 . . . 4 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → (𝑥𝐵𝑦𝐵𝑥𝑅𝑦))
98simp2d 1012 . . 3 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦𝐵)
108simp1d 1011 . . 3 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥𝐵)
11 erinxp.r . . . . 5 (𝜑𝑅 Er 𝐴)
1211adantr 276 . . . 4 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑅 Er 𝐴)
138simp3d 1013 . . . 4 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥𝑅𝑦)
1412, 13ersym 6571 . . 3 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦𝑅𝑥)
15 brinxp2 4711 . . 3 (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑦𝐵𝑥𝐵𝑦𝑅𝑥))
169, 10, 14, 15syl3anbrc 1183 . 2 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥)
1710adantrr 479 . . 3 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝐵)
18 simprr 531 . . . . 5 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)
19 brinxp2 4711 . . . . 5 (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ (𝑦𝐵𝑧𝐵𝑦𝑅𝑧))
2018, 19sylib 122 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → (𝑦𝐵𝑧𝐵𝑦𝑅𝑧))
2120simp2d 1012 . . 3 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑧𝐵)
2211adantr 276 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑅 Er 𝐴)
2313adantrr 479 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑦)
2420simp3d 1013 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦𝑅𝑧)
2522, 23, 24ertrd 6575 . . 3 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑧)
26 brinxp2 4711 . . 3 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ (𝑥𝐵𝑧𝐵𝑥𝑅𝑧))
2717, 21, 25, 26syl3anbrc 1183 . 2 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧)
2811adantr 276 . . . . . 6 ((𝜑𝑥𝐵) → 𝑅 Er 𝐴)
29 erinxp.a . . . . . . 7 (𝜑𝐵𝐴)
3029sselda 3170 . . . . . 6 ((𝜑𝑥𝐵) → 𝑥𝐴)
3128, 30erref 6579 . . . . 5 ((𝜑𝑥𝐵) → 𝑥𝑅𝑥)
3231ex 115 . . . 4 (𝜑 → (𝑥𝐵𝑥𝑅𝑥))
3332pm4.71rd 394 . . 3 (𝜑 → (𝑥𝐵 ↔ (𝑥𝑅𝑥𝑥𝐵)))
34 brin 4070 . . . 4 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥𝑥(𝐵 × 𝐵)𝑥))
35 brxp 4675 . . . . . 6 (𝑥(𝐵 × 𝐵)𝑥 ↔ (𝑥𝐵𝑥𝐵))
36 anidm 396 . . . . . 6 ((𝑥𝐵𝑥𝐵) ↔ 𝑥𝐵)
3735, 36bitri 184 . . . . 5 (𝑥(𝐵 × 𝐵)𝑥𝑥𝐵)
3837anbi2i 457 . . . 4 ((𝑥𝑅𝑥𝑥(𝐵 × 𝐵)𝑥) ↔ (𝑥𝑅𝑥𝑥𝐵))
3934, 38bitri 184 . . 3 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥𝑥𝐵))
4033, 39bitr4di 198 . 2 (𝜑 → (𝑥𝐵𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥))
415, 16, 27, 40iserd 6585 1 (𝜑 → (𝑅 ∩ (𝐵 × 𝐵)) Er 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980  wcel 2160  cin 3143  wss 3144   class class class wbr 4018   × cxp 4642  Rel wrel 4649   Er wer 6556
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 4192  ax-pr 4227
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-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-er 6559
This theorem is referenced by: (None)
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