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Theorem erinxp 6366
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 3221 . . . 4 (𝑅 ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
2 relxp 4547 . . . 4 Rel (𝐵 × 𝐵)
3 relss 4525 . . . 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 108 . . . . 5 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦)
7 brinxp2 4505 . . . . 5 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ↔ (𝑥𝐵𝑦𝐵𝑥𝑅𝑦))
86, 7sylib 120 . . . 4 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → (𝑥𝐵𝑦𝐵𝑥𝑅𝑦))
98simp2d 956 . . 3 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦𝐵)
108simp1d 955 . . 3 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥𝐵)
11 erinxp.r . . . . 5 (𝜑𝑅 Er 𝐴)
1211adantr 270 . . . 4 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑅 Er 𝐴)
138simp3d 957 . . . 4 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥𝑅𝑦)
1412, 13ersym 6304 . . 3 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦𝑅𝑥)
15 brinxp2 4505 . . 3 (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑦𝐵𝑥𝐵𝑦𝑅𝑥))
169, 10, 14, 15syl3anbrc 1127 . 2 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥)
1710adantrr 463 . . 3 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝐵)
18 simprr 499 . . . . 5 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)
19 brinxp2 4505 . . . . 5 (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ (𝑦𝐵𝑧𝐵𝑦𝑅𝑧))
2018, 19sylib 120 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → (𝑦𝐵𝑧𝐵𝑦𝑅𝑧))
2120simp2d 956 . . 3 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑧𝐵)
2211adantr 270 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑅 Er 𝐴)
2313adantrr 463 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑦)
2420simp3d 957 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦𝑅𝑧)
2522, 23, 24ertrd 6308 . . 3 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑧)
26 brinxp2 4505 . . 3 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ (𝑥𝐵𝑧𝐵𝑥𝑅𝑧))
2717, 21, 25, 26syl3anbrc 1127 . 2 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧)
2811adantr 270 . . . . . 6 ((𝜑𝑥𝐵) → 𝑅 Er 𝐴)
29 erinxp.a . . . . . . 7 (𝜑𝐵𝐴)
3029sselda 3025 . . . . . 6 ((𝜑𝑥𝐵) → 𝑥𝐴)
3128, 30erref 6312 . . . . 5 ((𝜑𝑥𝐵) → 𝑥𝑅𝑥)
3231ex 113 . . . 4 (𝜑 → (𝑥𝐵𝑥𝑅𝑥))
3332pm4.71rd 386 . . 3 (𝜑 → (𝑥𝐵 ↔ (𝑥𝑅𝑥𝑥𝐵)))
34 brin 3892 . . . 4 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥𝑥(𝐵 × 𝐵)𝑥))
35 brxp 4468 . . . . . 6 (𝑥(𝐵 × 𝐵)𝑥 ↔ (𝑥𝐵𝑥𝐵))
36 anidm 388 . . . . . 6 ((𝑥𝐵𝑥𝐵) ↔ 𝑥𝐵)
3735, 36bitri 182 . . . . 5 (𝑥(𝐵 × 𝐵)𝑥𝑥𝐵)
3837anbi2i 445 . . . 4 ((𝑥𝑅𝑥𝑥(𝐵 × 𝐵)𝑥) ↔ (𝑥𝑅𝑥𝑥𝐵))
3934, 38bitri 182 . . 3 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥𝑥𝐵))
4033, 39syl6bbr 196 . 2 (𝜑 → (𝑥𝐵𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥))
415, 16, 27, 40iserd 6318 1 (𝜑 → (𝑅 ∩ (𝐵 × 𝐵)) Er 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 924  wcel 1438  cin 2998  wss 2999   class class class wbr 3845   × cxp 4436  Rel wrel 4443   Er wer 6289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-er 6292
This theorem is referenced by: (None)
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