prproropf1olem0 - Mathbox for Alexander van der Vekens

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Theorem prproropf1olem0 48139

Description: Lemma 0 for prproropf1o 48144. 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) ∣ (1^st ‘𝑥)𝑅(2^nd ‘𝑥)} or even as 𝑂 = {𝑥 ∈ (𝑉 × 𝑉) ∣ ⟨(1^st ‘𝑥), (2^nd ‘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	⊢ (𝑊 ∈ 𝑂 ↔ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊)))

**Proof of Theorem prproropf1olem0**
Step	Hyp	Ref	Expression
1		prproropf1o.o	. . 3 ⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉))
2	1	eleq2i 2861	. 2 ⊢ (𝑊 ∈ 𝑂 ↔ 𝑊 ∈ (𝑅 ∩ (𝑉 × 𝑉)))
3		elin 3929	. 2 ⊢ (𝑊 ∈ (𝑅 ∩ (𝑉 × 𝑉)) ↔ (𝑊 ∈ 𝑅 ∧ 𝑊 ∈ (𝑉 × 𝑉)))
4		ancom 465	. . . 4 ⊢ ((𝑊 ∈ 𝑅 ∧ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉))) ↔ ((𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉)) ∧ 𝑊 ∈ 𝑅))
5		eleq1 2857	. . . . . . 7 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ 𝑅 ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ 𝑅))
6		df-br 5114	. . . . . . 7 ⊢ ((1^st ‘𝑊)𝑅(2^nd ‘𝑊) ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ 𝑅)
7	5, 6	bitr4di 292	. . . . . 6 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ 𝑅 ↔ (1^st ‘𝑊)𝑅(2^nd ‘𝑊)))
8	7	adantr 485	. . . . 5 ⊢ ((𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉)) → (𝑊 ∈ 𝑅 ↔ (1^st ‘𝑊)𝑅(2^nd ‘𝑊)))
9	8	pm5.32i 584	. . . 4 ⊢ (((𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉)) ∧ 𝑊 ∈ 𝑅) ↔ ((𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉)) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊)))
10	4, 9	bitri 278	. . 3 ⊢ ((𝑊 ∈ 𝑅 ∧ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉))) ↔ ((𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉)) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊)))
11		elxp6 8019	. . . 4 ⊢ (𝑊 ∈ (𝑉 × 𝑉) ↔ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉)))
12	11	anbi2i 634	. . 3 ⊢ ((𝑊 ∈ 𝑅 ∧ 𝑊 ∈ (𝑉 × 𝑉)) ↔ (𝑊 ∈ 𝑅 ∧ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉))))
13		df-3an 1103	. . 3 ⊢ ((𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊)) ↔ ((𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉)) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊)))
14	10, 12, 13	3bitr4i 306	. 2 ⊢ ((𝑊 ∈ 𝑅 ∧ 𝑊 ∈ (𝑉 × 𝑉)) ↔ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊)))
15	2, 3, 14	3bitri 300	1 ⊢ (𝑊 ∈ 𝑂 ↔ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉) ∧ (1^st ‘𝑊)𝑅(2^nd ‘𝑊)))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ∩ cin 3912 ⟨cop 4600 class class class wbr 5113 × cxp 5660 ‘cfv 6537 1^st c1st 7983 2^nd c2nd 7984

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5271 ax-pr 5405 ax-un 7733

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-rab 3424 df-v 3465 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-br 5114 df-opab 5178 df-mpt 5197 df-id 5557 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-iota 6493 df-fun 6539 df-fv 6545 df-1st 7985 df-2nd 7986

This theorem is referenced by: prproropf1olem1 48140 prproropf1olem3 48142 prproropf1o 48144

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