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Theorem asymref 4735
Description: Two ways of saying a relation is antisymmetric and reflexive. 𝑅 is the field of a relation by relfld 4871. (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 3790 . . . . . . . . . . 11 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
2 vex 2575 . . . . . . . . . . . 12 𝑥 ∈ V
3 vex 2575 . . . . . . . . . . . 12 𝑦 ∈ V
42, 3opeluu 4207 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → (𝑥 𝑅𝑦 𝑅))
51, 4sylbi 118 . . . . . . . . . 10 (𝑥𝑅𝑦 → (𝑥 𝑅𝑦 𝑅))
65simpld 109 . . . . . . . . 9 (𝑥𝑅𝑦𝑥 𝑅)
76adantr 265 . . . . . . . 8 ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 𝑅)
87pm4.71ri 378 . . . . . . 7 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥 𝑅 ∧ (𝑥𝑅𝑦𝑦𝑅𝑥)))
98bibi1i 221 . . . . . 6 (((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥 𝑅𝑥 = 𝑦)) ↔ ((𝑥 𝑅 ∧ (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ (𝑥 𝑅𝑥 = 𝑦)))
10 elin 3151 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
112, 3brcnv 4543 . . . . . . . . . 10 (𝑥𝑅𝑦𝑦𝑅𝑥)
12 df-br 3790 . . . . . . . . . 10 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
1311, 12bitr3i 179 . . . . . . . . 9 (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
141, 13anbi12i 441 . . . . . . . 8 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
1510, 14bitr4i 180 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥))
163opelres 4642 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 𝑅))
17 df-br 3790 . . . . . . . . . 10 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
183ideq 4513 . . . . . . . . . 10 (𝑥 I 𝑦𝑥 = 𝑦)
1917, 18bitr3i 179 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
2019anbi2ci 440 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 𝑅) ↔ (𝑥 𝑅𝑥 = 𝑦))
2116, 20bitri 177 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅) ↔ (𝑥 𝑅𝑥 = 𝑦))
2215, 21bibi12i 222 . . . . . 6 ((⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥 𝑅𝑥 = 𝑦)))
23 pm5.32 434 . . . . . 6 ((𝑥 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)) ↔ ((𝑥 𝑅 ∧ (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ (𝑥 𝑅𝑥 = 𝑦)))
249, 22, 233bitr4i 205 . . . . 5 ((⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ (𝑥 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
2524albii 1373 . . . 4 (∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ ∀𝑦(𝑥 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
26 19.21v 1767 . . . 4 (∀𝑦(𝑥 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)) ↔ (𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
2725, 26bitri 177 . . 3 (∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ (𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
2827albii 1373 . 2 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ ∀𝑥(𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
29 relcnv 4728 . . . 4 Rel 𝑅
30 relin2 4481 . . . 4 (Rel 𝑅 → Rel (𝑅𝑅))
3129, 30ax-mp 7 . . 3 Rel (𝑅𝑅)
32 relres 4664 . . 3 Rel ( I ↾ 𝑅)
33 eqrel 4454 . . 3 ((Rel (𝑅𝑅) ∧ Rel ( I ↾ 𝑅)) → ((𝑅𝑅) = ( I ↾ 𝑅) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅))))
3431, 32, 33mp2an 410 . 2 ((𝑅𝑅) = ( I ↾ 𝑅) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)))
35 df-ral 2326 . 2 (∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
3628, 34, 353bitr4i 205 1 ((𝑅𝑅) = ( I ↾ 𝑅) ↔ ∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1255   = wceq 1257  wcel 1407  wral 2321  cin 2941  cop 3403   cuni 3605   class class class wbr 3789   I cid 4050  ccnv 4369  cres 4372  Rel wrel 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-br 3790  df-opab 3844  df-id 4055  df-xp 4376  df-rel 4377  df-cnv 4378  df-res 4382
This theorem is referenced by: (None)
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