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 44062
 Description: Lemma 0 for prproropf1o 44067. 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) ∣ (1st ‘𝑥)𝑅(2nd ‘𝑥)} or even as 𝑂 = {𝑥 ∈ (𝑉 × 𝑉) ∣ ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ ∈ 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 2881 . 2 (𝑊𝑂𝑊 ∈ (𝑅 ∩ (𝑉 × 𝑉)))
3 elin 3897 . 2 (𝑊 ∈ (𝑅 ∩ (𝑉 × 𝑉)) ↔ (𝑊𝑅𝑊 ∈ (𝑉 × 𝑉)))
4 ancom 464 . . . 4 ((𝑊𝑅 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉))) ↔ ((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) ∧ 𝑊𝑅))
5 eleq1 2877 . . . . . . 7 (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ → (𝑊𝑅 ↔ ⟨(1st𝑊), (2nd𝑊)⟩ ∈ 𝑅))
6 df-br 5032 . . . . . . 7 ((1st𝑊)𝑅(2nd𝑊) ↔ ⟨(1st𝑊), (2nd𝑊)⟩ ∈ 𝑅)
75, 6bitr4di 292 . . . . . 6 (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ → (𝑊𝑅 ↔ (1st𝑊)𝑅(2nd𝑊)))
87adantr 484 . . . . 5 ((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) → (𝑊𝑅 ↔ (1st𝑊)𝑅(2nd𝑊)))
98pm5.32i 578 . . . 4 (((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) ∧ 𝑊𝑅) ↔ ((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) ∧ (1st𝑊)𝑅(2nd𝑊)))
104, 9bitri 278 . . 3 ((𝑊𝑅 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉))) ↔ ((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) ∧ (1st𝑊)𝑅(2nd𝑊)))
11 elxp6 7708 . . . 4 (𝑊 ∈ (𝑉 × 𝑉) ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)))
1211anbi2i 625 . . 3 ((𝑊𝑅𝑊 ∈ (𝑉 × 𝑉)) ↔ (𝑊𝑅 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉))))
13 df-3an 1086 . . 3 ((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊)) ↔ ((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) ∧ (1st𝑊)𝑅(2nd𝑊)))
1410, 12, 133bitr4i 306 . 2 ((𝑊𝑅𝑊 ∈ (𝑉 × 𝑉)) ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊)))
152, 3, 143bitri 300 1 (𝑊𝑂 ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ∩ cin 3880  ⟨cop 4531   class class class wbr 5031   × cxp 5518  ‘cfv 6325  1st c1st 7672  2nd c2nd 7673 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-iota 6284  df-fun 6327  df-fv 6333  df-1st 7674  df-2nd 7675 This theorem is referenced by:  prproropf1olem1  44063  prproropf1olem3  44065  prproropf1o  44067
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