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Theorem erinxp 7977
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 3982 . . . 4 (𝑅 ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
2 relxp 5267 . . . 4 Rel (𝐵 × 𝐵)
3 relss 5345 . . . 4 ((𝑅 ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵) → (Rel (𝐵 × 𝐵) → Rel (𝑅 ∩ (𝐵 × 𝐵))))
41, 2, 3mp2 9 . . 3 Rel (𝑅 ∩ (𝐵 × 𝐵))
54a1i 11 . 2 (𝜑 → Rel (𝑅 ∩ (𝐵 × 𝐵)))
6 simpr 471 . . . . 5 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦)
7 brinxp2 5319 . . . . 5 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ↔ (𝑥𝐵𝑦𝐵𝑥𝑅𝑦))
86, 7sylib 208 . . . 4 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → (𝑥𝐵𝑦𝐵𝑥𝑅𝑦))
98simp2d 1137 . . 3 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦𝐵)
108simp1d 1136 . . 3 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥𝐵)
11 erinxp.r . . . . 5 (𝜑𝑅 Er 𝐴)
1211adantr 466 . . . 4 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑅 Er 𝐴)
138simp3d 1138 . . . 4 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥𝑅𝑦)
1412, 13ersym 7912 . . 3 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦𝑅𝑥)
15 brinxp2 5319 . . 3 (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑦𝐵𝑥𝐵𝑦𝑅𝑥))
169, 10, 14, 15syl3anbrc 1428 . 2 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥)
1710adantrr 696 . . 3 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝐵)
18 simprr 756 . . . . 5 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)
19 brinxp2 5319 . . . . 5 (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ (𝑦𝐵𝑧𝐵𝑦𝑅𝑧))
2018, 19sylib 208 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → (𝑦𝐵𝑧𝐵𝑦𝑅𝑧))
2120simp2d 1137 . . 3 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑧𝐵)
2211adantr 466 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑅 Er 𝐴)
2313adantrr 696 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑦)
2420simp3d 1138 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦𝑅𝑧)
2522, 23, 24ertrd 7916 . . 3 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑧)
26 brinxp2 5319 . . 3 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ (𝑥𝐵𝑧𝐵𝑥𝑅𝑧))
2717, 21, 25, 26syl3anbrc 1428 . 2 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧)
2811adantr 466 . . . . . 6 ((𝜑𝑥𝐵) → 𝑅 Er 𝐴)
29 erinxp.a . . . . . . 7 (𝜑𝐵𝐴)
3029sselda 3752 . . . . . 6 ((𝜑𝑥𝐵) → 𝑥𝐴)
3128, 30erref 7920 . . . . 5 ((𝜑𝑥𝐵) → 𝑥𝑅𝑥)
3231ex 397 . . . 4 (𝜑 → (𝑥𝐵𝑥𝑅𝑥))
3332pm4.71rd 552 . . 3 (𝜑 → (𝑥𝐵 ↔ (𝑥𝑅𝑥𝑥𝐵)))
34 brin 4839 . . . 4 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥𝑥(𝐵 × 𝐵)𝑥))
35 brxp 5286 . . . . . 6 (𝑥(𝐵 × 𝐵)𝑥 ↔ (𝑥𝐵𝑥𝐵))
36 anidm 554 . . . . . 6 ((𝑥𝐵𝑥𝐵) ↔ 𝑥𝐵)
3735, 36bitri 264 . . . . 5 (𝑥(𝐵 × 𝐵)𝑥𝑥𝐵)
3837anbi2i 609 . . . 4 ((𝑥𝑅𝑥𝑥(𝐵 × 𝐵)𝑥) ↔ (𝑥𝑅𝑥𝑥𝐵))
3934, 38bitri 264 . . 3 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥𝑥𝐵))
4033, 39syl6bbr 278 . 2 (𝜑 → (𝑥𝐵𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥))
415, 16, 27, 40iserd 7926 1 (𝜑 → (𝑅 ∩ (𝐵 × 𝐵)) Er 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071  wcel 2145  cin 3722  wss 3723   class class class wbr 4787   × cxp 5248  Rel wrel 5255   Er wer 7897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-er 7900
This theorem is referenced by:  frgpuplem  18392  pi1buni  23059  pi1addf  23066  pi1addval  23067  pi1grplem  23068
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