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Mirrors > Home > MPE Home > Th. List > op1steq | Structured version Visualization version GIF version |
Description: Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.) |
Ref | Expression |
---|---|
op1steq | ⊢ (𝐴 ∈ (𝑉 × 𝑊) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = 〈𝐵, 𝑥〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpss 5705 | . . 3 ⊢ (𝑉 × 𝑊) ⊆ (V × V) | |
2 | 1 | sseli 3991 | . 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V)) |
3 | eqid 2735 | . . . . . 6 ⊢ (2nd ‘𝐴) = (2nd ‘𝐴) | |
4 | eqopi 8049 | . . . . . 6 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = (2nd ‘𝐴))) → 𝐴 = 〈𝐵, (2nd ‘𝐴)〉) | |
5 | 3, 4 | mpanr2 704 | . . . . 5 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = 𝐵) → 𝐴 = 〈𝐵, (2nd ‘𝐴)〉) |
6 | fvex 6920 | . . . . . 6 ⊢ (2nd ‘𝐴) ∈ V | |
7 | opeq2 4879 | . . . . . . 7 ⊢ (𝑥 = (2nd ‘𝐴) → 〈𝐵, 𝑥〉 = 〈𝐵, (2nd ‘𝐴)〉) | |
8 | 7 | eqeq2d 2746 | . . . . . 6 ⊢ (𝑥 = (2nd ‘𝐴) → (𝐴 = 〈𝐵, 𝑥〉 ↔ 𝐴 = 〈𝐵, (2nd ‘𝐴)〉)) |
9 | 6, 8 | spcev 3606 | . . . . 5 ⊢ (𝐴 = 〈𝐵, (2nd ‘𝐴)〉 → ∃𝑥 𝐴 = 〈𝐵, 𝑥〉) |
10 | 5, 9 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = 𝐵) → ∃𝑥 𝐴 = 〈𝐵, 𝑥〉) |
11 | 10 | ex 412 | . . 3 ⊢ (𝐴 ∈ (V × V) → ((1st ‘𝐴) = 𝐵 → ∃𝑥 𝐴 = 〈𝐵, 𝑥〉)) |
12 | eqop 8055 | . . . . 5 ⊢ (𝐴 ∈ (V × V) → (𝐴 = 〈𝐵, 𝑥〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝑥))) | |
13 | simpl 482 | . . . . 5 ⊢ (((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝑥) → (1st ‘𝐴) = 𝐵) | |
14 | 12, 13 | biimtrdi 253 | . . . 4 ⊢ (𝐴 ∈ (V × V) → (𝐴 = 〈𝐵, 𝑥〉 → (1st ‘𝐴) = 𝐵)) |
15 | 14 | exlimdv 1931 | . . 3 ⊢ (𝐴 ∈ (V × V) → (∃𝑥 𝐴 = 〈𝐵, 𝑥〉 → (1st ‘𝐴) = 𝐵)) |
16 | 11, 15 | impbid 212 | . 2 ⊢ (𝐴 ∈ (V × V) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = 〈𝐵, 𝑥〉)) |
17 | 2, 16 | syl 17 | 1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = 〈𝐵, 𝑥〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 〈cop 4637 × cxp 5687 ‘cfv 6563 1st c1st 8011 2nd c2nd 8012 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-1st 8013 df-2nd 8014 |
This theorem is referenced by: releldm2 8067 |
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