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Theorem asymref 5813
Description: Two ways of saying a relation is antisymmetric and reflexive. 𝑅 is the field of a relation by relfld 5961. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
asymref ((𝑅𝑅) = ( I ↾ 𝑅) ↔ ∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦))
Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem asymref
StepHypRef Expression
1 df-br 4926 . . . . . . . . . . 11 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
2 vex 3411 . . . . . . . . . . . 12 𝑥 ∈ V
3 vex 3411 . . . . . . . . . . . 12 𝑦 ∈ V
42, 3opeluu 5215 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → (𝑥 𝑅𝑦 𝑅))
51, 4sylbi 209 . . . . . . . . . 10 (𝑥𝑅𝑦 → (𝑥 𝑅𝑦 𝑅))
65simpld 487 . . . . . . . . 9 (𝑥𝑅𝑦𝑥 𝑅)
76adantr 473 . . . . . . . 8 ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 𝑅)
87pm4.71ri 553 . . . . . . 7 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥 𝑅 ∧ (𝑥𝑅𝑦𝑦𝑅𝑥)))
98bibi1i 331 . . . . . 6 (((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥 𝑅𝑥 = 𝑦)) ↔ ((𝑥 𝑅 ∧ (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ (𝑥 𝑅𝑥 = 𝑦)))
10 elin 4051 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
112, 3brcnv 5599 . . . . . . . . . 10 (𝑥𝑅𝑦𝑦𝑅𝑥)
12 df-br 4926 . . . . . . . . . 10 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
1311, 12bitr3i 269 . . . . . . . . 9 (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
141, 13anbi12i 618 . . . . . . . 8 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
1510, 14bitr4i 270 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥))
163opelresi 5700 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅) ↔ (𝑥 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
17 df-br 4926 . . . . . . . . . 10 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
183ideq 5569 . . . . . . . . . 10 (𝑥 I 𝑦𝑥 = 𝑦)
1917, 18bitr3i 269 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
2019anbi2i 614 . . . . . . . 8 ((𝑥 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 𝑅𝑥 = 𝑦))
2116, 20bitri 267 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅) ↔ (𝑥 𝑅𝑥 = 𝑦))
2215, 21bibi12i 332 . . . . . 6 ((⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥 𝑅𝑥 = 𝑦)))
23 pm5.32 566 . . . . . 6 ((𝑥 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)) ↔ ((𝑥 𝑅 ∧ (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ (𝑥 𝑅𝑥 = 𝑦)))
249, 22, 233bitr4i 295 . . . . 5 ((⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ (𝑥 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
2524albii 1783 . . . 4 (∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ ∀𝑦(𝑥 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
26 19.21v 1899 . . . 4 (∀𝑦(𝑥 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)) ↔ (𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
2725, 26bitri 267 . . 3 (∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ (𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
2827albii 1783 . 2 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ ∀𝑥(𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
29 relcnv 5804 . . . 4 Rel 𝑅
30 relin2 5532 . . . 4 (Rel 𝑅 → Rel (𝑅𝑅))
3129, 30ax-mp 5 . . 3 Rel (𝑅𝑅)
32 relres 5724 . . 3 Rel ( I ↾ 𝑅)
33 eqrel 5504 . . 3 ((Rel (𝑅𝑅) ∧ Rel ( I ↾ 𝑅)) → ((𝑅𝑅) = ( I ↾ 𝑅) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅))))
3431, 32, 33mp2an 680 . 2 ((𝑅𝑅) = ( I ↾ 𝑅) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)))
35 df-ral 3086 . 2 (∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
3628, 34, 353bitr4i 295 1 ((𝑅𝑅) = ( I ↾ 𝑅) ↔ ∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wal 1506   = wceq 1508  wcel 2051  wral 3081  cin 3821  cop 4441   cuni 4708   class class class wbr 4925   I cid 5307  ccnv 5402  cres 5405  Rel wrel 5408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-res 5415
This theorem is referenced by:  asymref2  5814  letsr  17707
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