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Theorem prproropf1olem0 43064
 Description: Lemma 0 for prproropf1o 43069. 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 2850 . 2 (𝑊𝑂𝑊 ∈ (𝑅 ∩ (𝑉 × 𝑉)))
3 elin 4051 . 2 (𝑊 ∈ (𝑅 ∩ (𝑉 × 𝑉)) ↔ (𝑊𝑅𝑊 ∈ (𝑉 × 𝑉)))
4 ancom 453 . . . 4 ((𝑊𝑅 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉))) ↔ ((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) ∧ 𝑊𝑅))
5 eleq1 2846 . . . . . . 7 (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ → (𝑊𝑅 ↔ ⟨(1st𝑊), (2nd𝑊)⟩ ∈ 𝑅))
6 df-br 4926 . . . . . . 7 ((1st𝑊)𝑅(2nd𝑊) ↔ ⟨(1st𝑊), (2nd𝑊)⟩ ∈ 𝑅)
75, 6syl6bbr 281 . . . . . 6 (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ → (𝑊𝑅 ↔ (1st𝑊)𝑅(2nd𝑊)))
87adantr 473 . . . . 5 ((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) → (𝑊𝑅 ↔ (1st𝑊)𝑅(2nd𝑊)))
98pm5.32i 567 . . . 4 (((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) ∧ 𝑊𝑅) ↔ ((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) ∧ (1st𝑊)𝑅(2nd𝑊)))
104, 9bitri 267 . . 3 ((𝑊𝑅 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉))) ↔ ((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) ∧ (1st𝑊)𝑅(2nd𝑊)))
11 elxp6 7533 . . . 4 (𝑊 ∈ (𝑉 × 𝑉) ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)))
1211anbi2i 614 . . 3 ((𝑊𝑅𝑊 ∈ (𝑉 × 𝑉)) ↔ (𝑊𝑅 ∧ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉))))
13 df-3an 1071 . . 3 ((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊)) ↔ ((𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉)) ∧ (1st𝑊)𝑅(2nd𝑊)))
1410, 12, 133bitr4i 295 . 2 ((𝑊𝑅𝑊 ∈ (𝑉 × 𝑉)) ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊)))
152, 3, 143bitri 289 1 (𝑊𝑂 ↔ (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ ∧ ((1st𝑊) ∈ 𝑉 ∧ (2nd𝑊) ∈ 𝑉) ∧ (1st𝑊)𝑅(2nd𝑊)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 387   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051   ∩ cin 3821  ⟨cop 4441   class class class wbr 4925   × cxp 5401  ‘cfv 6185  1st c1st 7497  2nd c2nd 7498 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-iota 6149  df-fun 6187  df-fv 6193  df-1st 7499  df-2nd 7500 This theorem is referenced by:  prproropf1olem1  43065  prproropf1olem3  43067  prproropf1o  43069
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