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Mirrors > Home > MPE Home > Th. List > Mathboxes > prproropf1olem0 | Structured version Visualization version GIF version |
Description: Lemma 0 for prproropf1o 45631. 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.) |
Ref | Expression |
---|---|
prproropf1o.o | ⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉)) |
Ref | Expression |
---|---|
prproropf1olem0 | ⊢ (𝑊 ∈ 𝑂 ↔ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prproropf1o.o | . . 3 ⊢ 𝑂 = (𝑅 ∩ (𝑉 × 𝑉)) | |
2 | 1 | eleq2i 2829 | . 2 ⊢ (𝑊 ∈ 𝑂 ↔ 𝑊 ∈ (𝑅 ∩ (𝑉 × 𝑉))) |
3 | elin 3924 | . 2 ⊢ (𝑊 ∈ (𝑅 ∩ (𝑉 × 𝑉)) ↔ (𝑊 ∈ 𝑅 ∧ 𝑊 ∈ (𝑉 × 𝑉))) | |
4 | ancom 461 | . . . 4 ⊢ ((𝑊 ∈ 𝑅 ∧ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉))) ↔ ((𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉)) ∧ 𝑊 ∈ 𝑅)) | |
5 | eleq1 2825 | . . . . . . 7 ⊢ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 → (𝑊 ∈ 𝑅 ↔ 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∈ 𝑅)) | |
6 | df-br 5104 | . . . . . . 7 ⊢ ((1st ‘𝑊)𝑅(2nd ‘𝑊) ↔ 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∈ 𝑅) | |
7 | 5, 6 | bitr4di 288 | . . . . . 6 ⊢ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 → (𝑊 ∈ 𝑅 ↔ (1st ‘𝑊)𝑅(2nd ‘𝑊))) |
8 | 7 | adantr 481 | . . . . 5 ⊢ ((𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉)) → (𝑊 ∈ 𝑅 ↔ (1st ‘𝑊)𝑅(2nd ‘𝑊))) |
9 | 8 | pm5.32i 575 | . . . 4 ⊢ (((𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉)) ∧ 𝑊 ∈ 𝑅) ↔ ((𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉)) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊))) |
10 | 4, 9 | bitri 274 | . . 3 ⊢ ((𝑊 ∈ 𝑅 ∧ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉))) ↔ ((𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉)) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊))) |
11 | elxp6 7951 | . . . 4 ⊢ (𝑊 ∈ (𝑉 × 𝑉) ↔ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉))) | |
12 | 11 | anbi2i 623 | . . 3 ⊢ ((𝑊 ∈ 𝑅 ∧ 𝑊 ∈ (𝑉 × 𝑉)) ↔ (𝑊 ∈ 𝑅 ∧ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉)))) |
13 | df-3an 1089 | . . 3 ⊢ ((𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊)) ↔ ((𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉)) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊))) | |
14 | 10, 12, 13 | 3bitr4i 302 | . 2 ⊢ ((𝑊 ∈ 𝑅 ∧ 𝑊 ∈ (𝑉 × 𝑉)) ↔ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊))) |
15 | 2, 3, 14 | 3bitri 296 | 1 ⊢ (𝑊 ∈ 𝑂 ↔ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝑉 ∧ (2nd ‘𝑊) ∈ 𝑉) ∧ (1st ‘𝑊)𝑅(2nd ‘𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∩ cin 3907 〈cop 4590 class class class wbr 5103 × cxp 5629 ‘cfv 6493 1st c1st 7915 2nd c2nd 7916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-iota 6445 df-fun 6495 df-fv 6501 df-1st 7917 df-2nd 7918 |
This theorem is referenced by: prproropf1olem1 45627 prproropf1olem3 45629 prproropf1o 45631 |
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