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Theorem asymref2 5731
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 5730 . 2 ((𝑅𝑅) = ( I ↾ 𝑅) ↔ ∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦))
2 albiim 1988 . . 3 (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦) ↔ (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥))))
32ralbii 3161 . 2 (∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦) ↔ ∀𝑥 𝑅(∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥))))
4 r19.26 3245 . . 3 (∀𝑥 𝑅(∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥))) ↔ (∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 𝑅𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥))))
5 ancom 453 . . 3 ((∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 𝑅𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥))) ↔ (∀𝑥 𝑅𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ ∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
6 equcom 2117 . . . . . . . 8 (𝑥 = 𝑦𝑦 = 𝑥)
76imbi1i 341 . . . . . . 7 ((𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑦𝑅𝑥)))
87albii 1915 . . . . . 6 (∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ ∀𝑦(𝑦 = 𝑥 → (𝑥𝑅𝑦𝑦𝑅𝑥)))
9 breq2 4847 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑥𝑅𝑥))
10 breq1 4846 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦𝑅𝑥𝑥𝑅𝑥))
119, 10anbi12d 625 . . . . . . . 8 (𝑦 = 𝑥 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑥𝑥𝑅𝑥)))
12 anidm 561 . . . . . . . 8 ((𝑥𝑅𝑥𝑥𝑅𝑥) ↔ 𝑥𝑅𝑥)
1311, 12syl6bb 279 . . . . . . 7 (𝑦 = 𝑥 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥𝑅𝑥))
1413equsalvw 2103 . . . . . 6 (∀𝑦(𝑦 = 𝑥 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ 𝑥𝑅𝑥)
158, 14bitri 267 . . . . 5 (∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ 𝑥𝑅𝑥)
1615ralbii 3161 . . . 4 (∀𝑥 𝑅𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ ∀𝑥 𝑅𝑥𝑅𝑥)
17 df-ral 3094 . . . . 5 (∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
18 df-br 4844 . . . . . . . . . . . . 13 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
19 vex 3388 . . . . . . . . . . . . . . 15 𝑥 ∈ V
20 vex 3388 . . . . . . . . . . . . . . 15 𝑦 ∈ V
2119, 20opeluu 5129 . . . . . . . . . . . . . 14 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → (𝑥 𝑅𝑦 𝑅))
2221simpld 489 . . . . . . . . . . . . 13 (⟨𝑥, 𝑦⟩ ∈ 𝑅𝑥 𝑅)
2318, 22sylbi 209 . . . . . . . . . . . 12 (𝑥𝑅𝑦𝑥 𝑅)
2423adantr 473 . . . . . . . . . . 11 ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 𝑅)
2524pm2.24d 149 . . . . . . . . . 10 ((𝑥𝑅𝑦𝑦𝑅𝑥) → (¬ 𝑥 𝑅𝑥 = 𝑦))
2625com12 32 . . . . . . . . 9 𝑥 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
2726alrimiv 2023 . . . . . . . 8 𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
28 id 22 . . . . . . . 8 (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
2927, 28ja 175 . . . . . . 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 201 . . . . . 6 ((𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)) ↔ ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3231albii 1915 . . . . 5 (∀𝑥(𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)) ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3317, 32bitri 267 . . . 4 (∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3416, 33anbi12i 621 . . 3 ((∀𝑥 𝑅𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ ∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)) ↔ (∀𝑥 𝑅𝑥𝑅𝑥 ∧ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
354, 5, 343bitri 289 . 2 (∀𝑥 𝑅(∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥))) ↔ (∀𝑥 𝑅𝑥𝑅𝑥 ∧ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
361, 3, 353bitri 289 1 ((𝑅𝑅) = ( I ↾ 𝑅) ↔ (∀𝑥 𝑅𝑥𝑅𝑥 ∧ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  wal 1651   = wceq 1653  wcel 2157  wral 3089  cin 3768  cop 4374   cuni 4628   class class class wbr 4843   I cid 5219  ccnv 5311  cres 5314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-res 5324
This theorem is referenced by:  pslem  17521  psss  17529
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