ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  erinxp GIF version

Theorem erinxp 6554
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 3328 . . . 4 (𝑅 ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
2 relxp 4695 . . . 4 Rel (𝐵 × 𝐵)
3 relss 4673 . . . 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 109 . . . . 5 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦)
7 brinxp2 4653 . . . . 5 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ↔ (𝑥𝐵𝑦𝐵𝑥𝑅𝑦))
86, 7sylib 121 . . . 4 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → (𝑥𝐵𝑦𝐵𝑥𝑅𝑦))
98simp2d 995 . . 3 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦𝐵)
108simp1d 994 . . 3 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥𝐵)
11 erinxp.r . . . . 5 (𝜑𝑅 Er 𝐴)
1211adantr 274 . . . 4 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑅 Er 𝐴)
138simp3d 996 . . . 4 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥𝑅𝑦)
1412, 13ersym 6492 . . 3 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦𝑅𝑥)
15 brinxp2 4653 . . 3 (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑦𝐵𝑥𝐵𝑦𝑅𝑥))
169, 10, 14, 15syl3anbrc 1166 . 2 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥)
1710adantrr 471 . . 3 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝐵)
18 simprr 522 . . . . 5 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)
19 brinxp2 4653 . . . . 5 (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ (𝑦𝐵𝑧𝐵𝑦𝑅𝑧))
2018, 19sylib 121 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → (𝑦𝐵𝑧𝐵𝑦𝑅𝑧))
2120simp2d 995 . . 3 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑧𝐵)
2211adantr 274 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑅 Er 𝐴)
2313adantrr 471 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑦)
2420simp3d 996 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦𝑅𝑧)
2522, 23, 24ertrd 6496 . . 3 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑧)
26 brinxp2 4653 . . 3 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ (𝑥𝐵𝑧𝐵𝑥𝑅𝑧))
2717, 21, 25, 26syl3anbrc 1166 . 2 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧)
2811adantr 274 . . . . . 6 ((𝜑𝑥𝐵) → 𝑅 Er 𝐴)
29 erinxp.a . . . . . . 7 (𝜑𝐵𝐴)
3029sselda 3128 . . . . . 6 ((𝜑𝑥𝐵) → 𝑥𝐴)
3128, 30erref 6500 . . . . 5 ((𝜑𝑥𝐵) → 𝑥𝑅𝑥)
3231ex 114 . . . 4 (𝜑 → (𝑥𝐵𝑥𝑅𝑥))
3332pm4.71rd 392 . . 3 (𝜑 → (𝑥𝐵 ↔ (𝑥𝑅𝑥𝑥𝐵)))
34 brin 4016 . . . 4 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥𝑥(𝐵 × 𝐵)𝑥))
35 brxp 4617 . . . . . 6 (𝑥(𝐵 × 𝐵)𝑥 ↔ (𝑥𝐵𝑥𝐵))
36 anidm 394 . . . . . 6 ((𝑥𝐵𝑥𝐵) ↔ 𝑥𝐵)
3735, 36bitri 183 . . . . 5 (𝑥(𝐵 × 𝐵)𝑥𝑥𝐵)
3837anbi2i 453 . . . 4 ((𝑥𝑅𝑥𝑥(𝐵 × 𝐵)𝑥) ↔ (𝑥𝑅𝑥𝑥𝐵))
3934, 38bitri 183 . . 3 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥𝑥𝐵))
4033, 39bitr4di 197 . 2 (𝜑 → (𝑥𝐵𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥))
415, 16, 27, 40iserd 6506 1 (𝜑 → (𝑅 ∩ (𝐵 × 𝐵)) Er 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 963  wcel 2128  cin 3101  wss 3102   class class class wbr 3965   × cxp 4584  Rel wrel 4591   Er wer 6477
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-er 6480
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator