prproropf1olem0 - Mathbox for Alexander van der Vekens

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Theorem prproropf1olem0 43658

Description: Lemma 0 for prproropf1o 43663. 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 2904	. 2 ⊢ (𝑊 ∈ 𝑂 ↔ 𝑊 ∈ (𝑅 ∩ (𝑉 × 𝑉)))
3		elin 4168	. 2 ⊢ (𝑊 ∈ (𝑅 ∩ (𝑉 × 𝑉)) ↔ (𝑊 ∈ 𝑅 ∧ 𝑊 ∈ (𝑉 × 𝑉)))
4		ancom 463	. . . 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 2900	. . . . . . 7 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ 𝑅 ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ 𝑅))
6		df-br 5059	. . . . . . 7 ⊢ ((1^st ‘𝑊)𝑅(2^nd ‘𝑊) ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ 𝑅)
7	5, 6	syl6bbr 291	. . . . . 6 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ 𝑅 ↔ (1^st ‘𝑊)𝑅(2^nd ‘𝑊)))
8	7	adantr 483	. . . . 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 577	. . . 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 277	. . 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 7717	. . . 4 ⊢ (𝑊 ∈ (𝑉 × 𝑉) ↔ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉)))
12	11	anbi2i 624	. . 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 1085	. . 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 305	. 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 299	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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 ⟨cop 4566 class class class wbr 5058 × cxp 5547 ‘cfv 6349 1^st c1st 7681 2^nd c2nd 7682

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fv 6357 df-1st 7683 df-2nd 7684

This theorem is referenced by: prproropf1olem1 43659 prproropf1olem3 43661 prproropf1o 43663

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