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Theorem asymref 4710
 Description: Two ways of saying a relation is antisymmetric and reflexive. ∪ ∪ 𝑅 is the field of a relation by relfld 4846. (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 3765 . . . . . . . . . . 11 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
2 vex 2560 . . . . . . . . . . . 12 𝑥 ∈ V
3 vex 2560 . . . . . . . . . . . 12 𝑦 ∈ V
42, 3opeluu 4182 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → (𝑥 𝑅𝑦 𝑅))
51, 4sylbi 114 . . . . . . . . . 10 (𝑥𝑅𝑦 → (𝑥 𝑅𝑦 𝑅))
65simpld 105 . . . . . . . . 9 (𝑥𝑅𝑦𝑥 𝑅)
76adantr 261 . . . . . . . 8 ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 𝑅)
87pm4.71ri 372 . . . . . . 7 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥 𝑅 ∧ (𝑥𝑅𝑦𝑦𝑅𝑥)))
98bibi1i 217 . . . . . 6 (((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥 𝑅𝑥 = 𝑦)) ↔ ((𝑥 𝑅 ∧ (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ (𝑥 𝑅𝑥 = 𝑦)))
10 elin 3126 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
112, 3brcnv 4518 . . . . . . . . . 10 (𝑥𝑅𝑦𝑦𝑅𝑥)
12 df-br 3765 . . . . . . . . . 10 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
1311, 12bitr3i 175 . . . . . . . . 9 (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
141, 13anbi12i 433 . . . . . . . 8 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
1510, 14bitr4i 176 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥))
163opelres 4617 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 𝑅))
17 df-br 3765 . . . . . . . . . 10 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
183ideq 4488 . . . . . . . . . 10 (𝑥 I 𝑦𝑥 = 𝑦)
1917, 18bitr3i 175 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
2019anbi2ci 432 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 𝑅) ↔ (𝑥 𝑅𝑥 = 𝑦))
2116, 20bitri 173 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅) ↔ (𝑥 𝑅𝑥 = 𝑦))
2215, 21bibi12i 218 . . . . . 6 ((⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥 𝑅𝑥 = 𝑦)))
23 pm5.32 426 . . . . . 6 ((𝑥 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)) ↔ ((𝑥 𝑅 ∧ (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ (𝑥 𝑅𝑥 = 𝑦)))
249, 22, 233bitr4i 201 . . . . 5 ((⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ (𝑥 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
2524albii 1359 . . . 4 (∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ ∀𝑦(𝑥 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
26 19.21v 1753 . . . 4 (∀𝑦(𝑥 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)) ↔ (𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
2725, 26bitri 173 . . 3 (∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ (𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
2827albii 1359 . 2 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)) ↔ ∀𝑥(𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
29 relcnv 4703 . . . 4 Rel 𝑅
30 relin2 4456 . . . 4 (Rel 𝑅 → Rel (𝑅𝑅))
3129, 30ax-mp 7 . . 3 Rel (𝑅𝑅)
32 relres 4639 . . 3 Rel ( I ↾ 𝑅)
33 eqrel 4429 . . 3 ((Rel (𝑅𝑅) ∧ Rel ( I ↾ 𝑅)) → ((𝑅𝑅) = ( I ↾ 𝑅) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅))))
3431, 32, 33mp2an 402 . 2 ((𝑅𝑅) = ( I ↾ 𝑅) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ⟨𝑥, 𝑦⟩ ∈ ( I ↾ 𝑅)))
35 df-ral 2311 . 2 (∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦)))
3628, 34, 353bitr4i 201 1 ((𝑅𝑅) = ( I ↾ 𝑅) ↔ ∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241   = wceq 1243   ∈ wcel 1393  ∀wral 2306   ∩ cin 2916  ⟨cop 3378  ∪ cuni 3580   class class class wbr 3764   I cid 4025  ◡ccnv 4344   ↾ cres 4347  Rel wrel 4350 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-res 4357 This theorem is referenced by: (None)
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