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Theorem erinxp 8784
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 5814 . . 3 Rel (𝑅 ∩ (𝐵 × 𝐵))
21a1i 11 . 2 (𝜑 → Rel (𝑅 ∩ (𝐵 × 𝐵)))
3 simpr 485 . . . . 5 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦)
4 brinxp2 5753 . . . . 5 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦 ↔ ((𝑥𝐵𝑦𝐵) ∧ 𝑥𝑅𝑦))
53, 4sylib 217 . . . 4 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → ((𝑥𝐵𝑦𝐵) ∧ 𝑥𝑅𝑦))
65simplrd 768 . . 3 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦𝐵)
75simplld 766 . . 3 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥𝐵)
8 erinxp.r . . . . 5 (𝜑𝑅 Er 𝐴)
98adantr 481 . . . 4 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑅 Er 𝐴)
105simprd 496 . . . 4 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑥𝑅𝑦)
119, 10ersym 8714 . . 3 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦𝑅𝑥)
12 brinxp2 5753 . . 3 (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ ((𝑦𝐵𝑥𝐵) ∧ 𝑦𝑅𝑥))
136, 7, 11, 12syl21anbrc 1344 . 2 ((𝜑𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑥)
147adantrr 715 . . 3 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝐵)
15 simprr 771 . . . . 5 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)
16 brinxp2 5753 . . . . 5 (𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ ((𝑦𝐵𝑧𝐵) ∧ 𝑦𝑅𝑧))
1715, 16sylib 217 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → ((𝑦𝐵𝑧𝐵) ∧ 𝑦𝑅𝑧))
1817simplrd 768 . . 3 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑧𝐵)
198adantr 481 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑅 Er 𝐴)
2010adantrr 715 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑦)
2117simprd 496 . . . 4 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑦𝑅𝑧)
2219, 20, 21ertrd 8718 . . 3 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥𝑅𝑧)
23 brinxp2 5753 . . 3 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧 ↔ ((𝑥𝐵𝑧𝐵) ∧ 𝑥𝑅𝑧))
2414, 18, 22, 23syl21anbrc 1344 . 2 ((𝜑 ∧ (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑦𝑦(𝑅 ∩ (𝐵 × 𝐵))𝑧)) → 𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑧)
258adantr 481 . . . . . 6 ((𝜑𝑥𝐵) → 𝑅 Er 𝐴)
26 erinxp.a . . . . . . 7 (𝜑𝐵𝐴)
2726sselda 3982 . . . . . 6 ((𝜑𝑥𝐵) → 𝑥𝐴)
2825, 27erref 8722 . . . . 5 ((𝜑𝑥𝐵) → 𝑥𝑅𝑥)
2928ex 413 . . . 4 (𝜑 → (𝑥𝐵𝑥𝑅𝑥))
3029pm4.71rd 563 . . 3 (𝜑 → (𝑥𝐵 ↔ (𝑥𝑅𝑥𝑥𝐵)))
31 brin 5200 . . . 4 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥𝑥(𝐵 × 𝐵)𝑥))
32 brxp 5725 . . . . . 6 (𝑥(𝐵 × 𝐵)𝑥 ↔ (𝑥𝐵𝑥𝐵))
33 anidm 565 . . . . . 6 ((𝑥𝐵𝑥𝐵) ↔ 𝑥𝐵)
3432, 33bitri 274 . . . . 5 (𝑥(𝐵 × 𝐵)𝑥𝑥𝐵)
3534anbi2i 623 . . . 4 ((𝑥𝑅𝑥𝑥(𝐵 × 𝐵)𝑥) ↔ (𝑥𝑅𝑥𝑥𝐵))
3631, 35bitri 274 . . 3 (𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥 ↔ (𝑥𝑅𝑥𝑥𝐵))
3730, 36bitr4di 288 . 2 (𝜑 → (𝑥𝐵𝑥(𝑅 ∩ (𝐵 × 𝐵))𝑥))
382, 13, 24, 37iserd 8728 1 (𝜑 → (𝑅 ∩ (𝐵 × 𝐵)) Er 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  cin 3947  wss 3948   class class class wbr 5148   × cxp 5674  Rel wrel 5681   Er wer 8699
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-er 8702
This theorem is referenced by:  frgpuplem  19639  pi1buni  24555  pi1addf  24562  pi1addval  24563  pi1grplem  24564
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