prproropf1olem0 - Mathbox for Alexander van der Vekens

	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 43734

Description: Lemma 0 for prproropf1o 43739. 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 2903	. 2 ⊢ (𝑊 ∈ 𝑂 ↔ 𝑊 ∈ (𝑅 ∩ (𝑉 × 𝑉)))
3		elin 4162	. 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 2899	. . . . . . 7 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ 𝑅 ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ 𝑅))
6		df-br 5060	. . . . . . 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 7716	. . . 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 1084	. . 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 1082 = wceq 1536 ∈ wcel 2113 ∩ cin 3928 ⟨cop 4566 class class class wbr 5059 × cxp 5546 ‘cfv 6348 1^st c1st 7680 2^nd c2nd 7681

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-1st 7682 df-2nd 7683

This theorem is referenced by: prproropf1olem1 43735 prproropf1olem3 43737 prproropf1o 43739

W3C validator