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Theorem asymref2 5976
Description: Two ways of saying a relation is antisymmetric and reflexive. (Contributed by NM, 6-May-2008.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
Assertion
Ref Expression
asymref2 ((𝑅𝑅) = ( I ↾ 𝑅) ↔ (∀𝑥 𝑅𝑥𝑅𝑥 ∧ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem asymref2
StepHypRef Expression
1 asymref 5975 . 2 ((𝑅𝑅) = ( I ↾ 𝑅) ↔ ∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦))
2 albiim 1886 . . 3 (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦) ↔ (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥))))
32ralbii 3165 . 2 (∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦) ↔ ∀𝑥 𝑅(∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥))))
4 r19.26 3170 . . 3 (∀𝑥 𝑅(∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥))) ↔ (∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 𝑅𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥))))
5 ancom 463 . . 3 ((∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 𝑅𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥))) ↔ (∀𝑥 𝑅𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ ∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
6 equcom 2021 . . . . . . . 8 (𝑥 = 𝑦𝑦 = 𝑥)
76imbi1i 352 . . . . . . 7 ((𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑦𝑅𝑥)))
87albii 1816 . . . . . 6 (∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ ∀𝑦(𝑦 = 𝑥 → (𝑥𝑅𝑦𝑦𝑅𝑥)))
9 breq2 5069 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑥𝑅𝑥))
10 breq1 5068 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦𝑅𝑥𝑥𝑅𝑥))
119, 10anbi12d 632 . . . . . . . 8 (𝑦 = 𝑥 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑥𝑥𝑅𝑥)))
12 anidm 567 . . . . . . . 8 ((𝑥𝑅𝑥𝑥𝑅𝑥) ↔ 𝑥𝑅𝑥)
1311, 12syl6bb 289 . . . . . . 7 (𝑦 = 𝑥 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥𝑅𝑥))
1413equsalvw 2006 . . . . . 6 (∀𝑦(𝑦 = 𝑥 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ 𝑥𝑅𝑥)
158, 14bitri 277 . . . . 5 (∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ 𝑥𝑅𝑥)
1615ralbii 3165 . . . 4 (∀𝑥 𝑅𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ ∀𝑥 𝑅𝑥𝑅𝑥)
17 df-ral 3143 . . . . 5 (∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
18 df-br 5066 . . . . . . . . . . . . 13 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
19 vex 3497 . . . . . . . . . . . . . . 15 𝑥 ∈ V
20 vex 3497 . . . . . . . . . . . . . . 15 𝑦 ∈ V
2119, 20opeluu 5361 . . . . . . . . . . . . . 14 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → (𝑥 𝑅𝑦 𝑅))
2221simpld 497 . . . . . . . . . . . . 13 (⟨𝑥, 𝑦⟩ ∈ 𝑅𝑥 𝑅)
2318, 22sylbi 219 . . . . . . . . . . . 12 (𝑥𝑅𝑦𝑥 𝑅)
2423adantr 483 . . . . . . . . . . 11 ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 𝑅)
2524pm2.24d 154 . . . . . . . . . 10 ((𝑥𝑅𝑦𝑦𝑅𝑥) → (¬ 𝑥 𝑅𝑥 = 𝑦))
2625com12 32 . . . . . . . . 9 𝑥 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
2726alrimiv 1924 . . . . . . . 8 𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
28 id 22 . . . . . . . 8 (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
2927, 28ja 188 . . . . . . 7 ((𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)) → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
30 ax-1 6 . . . . . . 7 (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) → (𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
3129, 30impbii 211 . . . . . 6 ((𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)) ↔ ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3231albii 1816 . . . . 5 (∀𝑥(𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)) ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3317, 32bitri 277 . . . 4 (∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3416, 33anbi12i 628 . . 3 ((∀𝑥 𝑅𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ ∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)) ↔ (∀𝑥 𝑅𝑥𝑅𝑥 ∧ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
354, 5, 343bitri 299 . 2 (∀𝑥 𝑅(∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥))) ↔ (∀𝑥 𝑅𝑥𝑅𝑥 ∧ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
361, 3, 353bitri 299 1 ((𝑅𝑅) = ( I ↾ 𝑅) ↔ (∀𝑥 𝑅𝑥𝑅𝑥 ∧ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wal 1531   = wceq 1533  wcel 2110  wral 3138  cin 3934  cop 4572   cuni 4837   class class class wbr 5065   I cid 5458  ccnv 5553  cres 5556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-res 5566
This theorem is referenced by:  pslem  17815  psss  17823
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