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Theorem asymref2 6022
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 6021 . 2 ((𝑅𝑅) = ( I ↾ 𝑅) ↔ ∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦))
2 albiim 1892 . . 3 (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦) ↔ (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥))))
32ralbii 3092 . 2 (∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦) ↔ ∀𝑥 𝑅(∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥))))
4 r19.26 3095 . . 3 (∀𝑥 𝑅(∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥))) ↔ (∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 𝑅𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥))))
5 ancom 461 . . 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 350 . . . . . . 7 ((𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑦𝑅𝑥)))
87albii 1822 . . . . . 6 (∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ ∀𝑦(𝑦 = 𝑥 → (𝑥𝑅𝑦𝑦𝑅𝑥)))
9 breq2 5078 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑥𝑅𝑥))
10 breq1 5077 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦𝑅𝑥𝑥𝑅𝑥))
119, 10anbi12d 631 . . . . . . . 8 (𝑦 = 𝑥 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑥𝑥𝑅𝑥)))
12 anidm 565 . . . . . . . 8 ((𝑥𝑅𝑥𝑥𝑅𝑥) ↔ 𝑥𝑅𝑥)
1311, 12bitrdi 287 . . . . . . 7 (𝑦 = 𝑥 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥𝑅𝑥))
1413equsalvw 2007 . . . . . 6 (∀𝑦(𝑦 = 𝑥 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ 𝑥𝑅𝑥)
158, 14bitri 274 . . . . 5 (∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ 𝑥𝑅𝑥)
1615ralbii 3092 . . . 4 (∀𝑥 𝑅𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ↔ ∀𝑥 𝑅𝑥𝑅𝑥)
17 df-ral 3069 . . . . 5 (∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
18 df-br 5075 . . . . . . . . . . . . 13 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
19 vex 3436 . . . . . . . . . . . . . . 15 𝑥 ∈ V
20 vex 3436 . . . . . . . . . . . . . . 15 𝑦 ∈ V
2119, 20opeluu 5385 . . . . . . . . . . . . . 14 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → (𝑥 𝑅𝑦 𝑅))
2221simpld 495 . . . . . . . . . . . . 13 (⟨𝑥, 𝑦⟩ ∈ 𝑅𝑥 𝑅)
2318, 22sylbi 216 . . . . . . . . . . . 12 (𝑥𝑅𝑦𝑥 𝑅)
2423adantr 481 . . . . . . . . . . 11 ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 𝑅)
2524pm2.24d 151 . . . . . . . . . 10 ((𝑥𝑅𝑦𝑦𝑅𝑥) → (¬ 𝑥 𝑅𝑥 = 𝑦))
2625com12 32 . . . . . . . . 9 𝑥 𝑅 → ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
2726alrimiv 1930 . . . . . . . 8 𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
28 id 22 . . . . . . . 8 (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
2927, 28ja 186 . . . . . . 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 208 . . . . . 6 ((𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)) ↔ ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3231albii 1822 . . . . 5 (∀𝑥(𝑥 𝑅 → ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)) ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3317, 32bitri 274 . . . 4 (∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3416, 33anbi12i 627 . . 3 ((∀𝑥 𝑅𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ ∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)) ↔ (∀𝑥 𝑅𝑥𝑅𝑥 ∧ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
354, 5, 343bitri 297 . 2 (∀𝑥 𝑅(∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦) ∧ ∀𝑦(𝑥 = 𝑦 → (𝑥𝑅𝑦𝑦𝑅𝑥))) ↔ (∀𝑥 𝑅𝑥𝑅𝑥 ∧ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
361, 3, 353bitri 297 1 ((𝑅𝑅) = ( I ↾ 𝑅) ↔ (∀𝑥 𝑅𝑥𝑅𝑥 ∧ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wal 1537   = wceq 1539  wcel 2106  wral 3064  cin 3886  cop 4567   cuni 4839   class class class wbr 5074   I cid 5488  ccnv 5588  cres 5591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-res 5601
This theorem is referenced by:  pslem  18290  psss  18298
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