Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prproropf1olem0 Structured version   Visualization version   GIF version

Theorem prproropf1olem0 45626
Description: Lemma 0 for prproropf1o 45631. Remark: 𝑂, the set of ordered ordered pairs, i.e., ordered pairs in which the first component is less than the second component, can alternatively be written as 𝑂 = {𝑥 ∈ (𝑉 × 𝑉) ∣ (1st𝑥)𝑅(2nd𝑥)} or even as 𝑂 = {𝑥 ∈ (𝑉 × 𝑉) ∣ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑅}, by which the relationship between ordered and unordered pair is immediately visible. (Contributed by AV, 18-Mar-2023.)
Hypothesis
Ref Expression
prproropf1o.o 𝑂 = (𝑅 ∩ (𝑉 × 𝑉))
Assertion
Ref Expression
prproropf1olem0 (𝑊𝑂 ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊)))

Proof of Theorem prproropf1olem0
StepHypRef Expression
1 prproropf1o.o . . 3 𝑂 = (𝑅 ∩ (𝑉 × 𝑉))
21eleq2i 2829 . 2 (𝑊𝑂𝑊 ∈ (𝑅 ∩ (𝑉 × 𝑉)))
3 elin 3924 . 2 (𝑊 ∈ (𝑅 ∩ (𝑉 × 𝑉)) ↔ (𝑊𝑅𝑊 ∈ (𝑉 × 𝑉)))
4 ancom 461 . . . 4 ((𝑊𝑅 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉))) ↔ ((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) ∧ 𝑊𝑅))
5 eleq1 2825 . . . . . . 7 (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ → (𝑊𝑅 ↔ ⟨(1st𝑊), (2nd𝑊)⟩ ∈ 𝑅))
6 df-br 5104 . . . . . . 7 ((1st𝑊)𝑅(2nd𝑊) ↔ ⟨(1st𝑊), (2nd𝑊)⟩ ∈ 𝑅)
75, 6bitr4di 288 . . . . . 6 (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ → (𝑊𝑅 ↔ (1st𝑊)𝑅(2nd𝑊)))
87adantr 481 . . . . 5 ((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) → (𝑊𝑅 ↔ (1st𝑊)𝑅(2nd𝑊)))
98pm5.32i 575 . . . 4 (((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) ∧ 𝑊𝑅) ↔ ((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) ∧ (1st𝑊)𝑅(2nd𝑊)))
104, 9bitri 274 . . 3 ((𝑊𝑅 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉))) ↔ ((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) ∧ (1st𝑊)𝑅(2nd𝑊)))
11 elxp6 7951 . . . 4 (𝑊 ∈ (𝑉 × 𝑉) ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)))
1211anbi2i 623 . . 3 ((𝑊𝑅𝑊 ∈ (𝑉 × 𝑉)) ↔ (𝑊𝑅 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉))))
13 df-3an 1089 . . 3 ((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊)) ↔ ((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) ∧ (1st𝑊)𝑅(2nd𝑊)))
1410, 12, 133bitr4i 302 . 2 ((𝑊𝑅𝑊 ∈ (𝑉 × 𝑉)) ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊)))
152, 3, 143bitri 296 1 (𝑊𝑂 ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  cin 3907  cop 4590   class class class wbr 5103   × cxp 5629  cfv 6493  1st c1st 7915  2nd c2nd 7916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6445  df-fun 6495  df-fv 6501  df-1st 7917  df-2nd 7918
This theorem is referenced by:  prproropf1olem1  45627  prproropf1olem3  45629  prproropf1o  45631
  Copyright terms: Public domain W3C validator