prproropf1olem0 - Mathbox for Alexander van der Vekens

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Theorem prproropf1olem0 47516

Description: Lemma 0 for prproropf1o 47521. 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 2826	. 2 ⊢ (𝑊 ∈ 𝑂 ↔ 𝑊 ∈ (𝑅 ∩ (𝑉 × 𝑉)))
3		elin 3942	. 2 ⊢ (𝑊 ∈ (𝑅 ∩ (𝑉 × 𝑉)) ↔ (𝑊 ∈ 𝑅 ∧ 𝑊 ∈ (𝑉 × 𝑉)))
4		ancom 460	. . . 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 2822	. . . . . . 7 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ 𝑅 ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ 𝑅))
6		df-br 5120	. . . . . . 7 ⊢ ((1^st ‘𝑊)𝑅(2^nd ‘𝑊) ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ 𝑅)
7	5, 6	bitr4di 289	. . . . . 6 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ 𝑅 ↔ (1^st ‘𝑊)𝑅(2^nd ‘𝑊)))
8	7	adantr 480	. . . . 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 574	. . . 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 275	. . 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 8022	. . . 4 ⊢ (𝑊 ∈ (𝑉 × 𝑉) ↔ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝑉 ∧ (2^nd ‘𝑊) ∈ 𝑉)))
12	11	anbi2i 623	. . 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 1088	. . 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 303	. 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 297	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ∩ cin 3925 ⟨cop 4607 class class class wbr 5119 × cxp 5652 ‘cfv 6531 1^st c1st 7986 2^nd c2nd 7987

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pr 5402 ax-un 7729

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-iota 6484 df-fun 6533 df-fv 6539 df-1st 7988 df-2nd 7989

This theorem is referenced by: prproropf1olem1 47517 prproropf1olem3 47519 prproropf1o 47521

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