Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  erinxp Structured version   Visualization version   GIF version

Theorem erinxp 8358
 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 relinxp 5655 . . 3 Rel (𝑅 ∩ (𝐵 × 𝐵))
21a1i 11 . 2 (𝜑 → Rel (𝑅 ∩ (𝐵 × 𝐵)))
3 simpr 488 . . . . 5 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦)
4 brinxp2 5597 . . . . 5 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ↔ ((𝑥𝐵𝑦𝐵) ∧ 𝑥𝑅𝑦))
53, 4sylib 221 . . . 4 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → ((𝑥𝐵𝑦𝐵) ∧ 𝑥𝑅𝑦))
65simplrd 769 . . 3 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦𝐵)
75simplld 767 . . 3 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥𝐵)
8 erinxp.r . . . . 5 (𝜑𝑅 Er 𝐴)
98adantr 484 . . . 4 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑅 Er 𝐴)
105simprd 499 . . . 4 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥𝑅𝑦)
119, 10ersym 8288 . . 3 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦𝑅𝑥)
12 brinxp2 5597 . . 3 (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ ((𝑦𝐵𝑥𝐵) ∧ 𝑦𝑅𝑥))
136, 7, 11, 12syl21anbrc 1341 . 2 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥)
147adantrr 716 . . 3 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝐵)
15 simprr 772 . . . . 5 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)
16 brinxp2 5597 . . . . 5 (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ ((𝑦𝐵𝑧𝐵) ∧ 𝑦𝑅𝑧))
1715, 16sylib 221 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → ((𝑦𝐵𝑧𝐵) ∧ 𝑦𝑅𝑧))
1817simplrd 769 . . 3 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑧𝐵)
198adantr 484 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑅 Er 𝐴)
2010adantrr 716 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑦)
2117simprd 499 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦𝑅𝑧)
2219, 20, 21ertrd 8292 . . 3 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑧)
23 brinxp2 5597 . . 3 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ ((𝑥𝐵𝑧𝐵) ∧ 𝑥𝑅𝑧))
2414, 18, 22, 23syl21anbrc 1341 . 2 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧)
258adantr 484 . . . . . 6 ((𝜑𝑥𝐵) → 𝑅 Er 𝐴)
26 erinxp.a . . . . . . 7 (𝜑𝐵𝐴)
2726sselda 3918 . . . . . 6 ((𝜑𝑥𝐵) → 𝑥𝐴)
2825, 27erref 8296 . . . . 5 ((𝜑𝑥𝐵) → 𝑥𝑅𝑥)
2928ex 416 . . . 4 (𝜑 → (𝑥𝐵𝑥𝑅𝑥))
3029pm4.71rd 566 . . 3 (𝜑 → (𝑥𝐵 ↔ (𝑥𝑅𝑥𝑥𝐵)))
31 brin 5085 . . . 4 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥𝑥(𝐵 × 𝐵)𝑥))
32 brxp 5569 . . . . . 6 (𝑥(𝐵 × 𝐵)𝑥 ↔ (𝑥𝐵𝑥𝐵))
33 anidm 568 . . . . . 6 ((𝑥𝐵𝑥𝐵) ↔ 𝑥𝐵)
3432, 33bitri 278 . . . . 5 (𝑥(𝐵 × 𝐵)𝑥𝑥𝐵)
3534anbi2i 625 . . . 4 ((𝑥𝑅𝑥𝑥(𝐵 × 𝐵)𝑥) ↔ (𝑥𝑅𝑥𝑥𝐵))
3631, 35bitri 278 . . 3 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥𝑥𝐵))
3730, 36syl6bbr 292 . 2 (𝜑 → (𝑥𝐵𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥))
382, 13, 24, 37iserd 8302 1 (𝜑 → (𝑅 ∩ (𝐵 × 𝐵)) Er 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2112   ∩ cin 3883   ⊆ wss 3884   class class class wbr 5033   × cxp 5521  Rel wrel 5528   Er wer 8273 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-er 8276 This theorem is referenced by:  frgpuplem  18894  pi1buni  23649  pi1addf  23656  pi1addval  23657  pi1grplem  23658
 Copyright terms: Public domain W3C validator