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Theorem asymref 4734
Description: Two ways of saying a relation is antisymmetric and reflexive. 𝑅 is the field of a relation by relfld 4870. (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 3788 . . . . . . . . . . 11 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
2 vex 2605 . . . . . . . . . . . 12 𝑥 ∈ V
3 vex 2605 . . . . . . . . . . . 12 𝑦 ∈ V
42, 3opeluu 4202 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → (𝑥 𝑅𝑦 𝑅))
51, 4sylbi 119 . . . . . . . . . 10 (𝑥𝑅𝑦 → (𝑥 𝑅𝑦 𝑅))
65simpld 110 . . . . . . . . 9 (𝑥𝑅𝑦𝑥 𝑅)
76adantr 270 . . . . . . . 8 ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 𝑅)
87pm4.71ri 384 . . . . . . 7 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥 𝑅 ∧ (𝑥𝑅𝑦𝑦𝑅𝑥)))
98bibi1i 226 . . . . . 6 (((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥 𝑅𝑥 = 𝑦)) ↔ ((𝑥 𝑅 ∧ (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ (𝑥 𝑅𝑥 = 𝑦)))
10 elin 3156 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
112, 3brcnv 4540 . . . . . . . . . 10 (𝑥𝑅𝑦𝑦𝑅𝑥)
12 df-br 3788 . . . . . . . . . 10 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
1311, 12bitr3i 184 . . . . . . . . 9 (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
141, 13anbi12i 448 . . . . . . . 8 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
1510, 14bitr4i 185 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥))
163opelres 4639 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 𝑅))
17 df-br 3788 . . . . . . . . . 10 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
183ideq 4510 . . . . . . . . . 10 (𝑥 I 𝑦𝑥 = 𝑦)
1917, 18bitr3i 184 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
2019anbi2ci 447 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 𝑅) ↔ (𝑥 𝑅𝑥 = 𝑦))
2116, 20bitri 182 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅) ↔ (𝑥 𝑅𝑥 = 𝑦))
2215, 21bibi12i 227 . . . . . 6 ((⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥 𝑅𝑥 = 𝑦)))
23 pm5.32 441 . . . . . 6 ((𝑥 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)) ↔ ((𝑥 𝑅 ∧ (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ (𝑥 𝑅𝑥 = 𝑦)))
249, 22, 233bitr4i 210 . . . . 5 ((⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ (𝑥 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
2524albii 1400 . . . 4 (∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ ∀𝑦(𝑥 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
26 19.21v 1795 . . . 4 (∀𝑦(𝑥 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)) ↔ (𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
2725, 26bitri 182 . . 3 (∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ (𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
2827albii 1400 . 2 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ ∀𝑥(𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
29 relcnv 4727 . . . 4 Rel 𝑅
30 relin2 4478 . . . 4 (Rel 𝑅 → Rel (𝑅𝑅))
3129, 30ax-mp 7 . . 3 Rel (𝑅𝑅)
32 relres 4661 . . 3 Rel ( I ↾ 𝑅)
33 eqrel 4449 . . 3 ((Rel (𝑅𝑅) ∧ Rel ( I ↾ 𝑅)) → ((𝑅𝑅) = ( I ↾ 𝑅) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅))))
3431, 32, 33mp2an 417 . 2 ((𝑅𝑅) = ( I ↾ 𝑅) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)))
35 df-ral 2354 . 2 (∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
3628, 34, 353bitr4i 210 1 ((𝑅𝑅) = ( I ↾ 𝑅) ↔ ∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283   = wceq 1285  wcel 1434  wral 2349  cin 2973  cop 3403   cuni 3603   class class class wbr 3787   I cid 4045  ccnv 4364  cres 4367  Rel wrel 4370
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-res 4377
This theorem is referenced by: (None)
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