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Mirrors > Home > ILE Home > Th. List > op1steq | 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 4504 | . . 3 ⊢ (𝑉 × 𝑊) ⊆ (V × V) | |
2 | 1 | sseli 3006 | . 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V)) |
3 | eqid 2083 | . . . . . 6 ⊢ (2nd ‘𝐴) = (2nd ‘𝐴) | |
4 | eqopi 5877 | . . . . . 6 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = (2nd ‘𝐴))) → 𝐴 = 〈𝐵, (2nd ‘𝐴)〉) | |
5 | 3, 4 | mpanr2 429 | . . . . 5 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = 𝐵) → 𝐴 = 〈𝐵, (2nd ‘𝐴)〉) |
6 | 2ndexg 5874 | . . . . . . 7 ⊢ (𝐴 ∈ (V × V) → (2nd ‘𝐴) ∈ V) | |
7 | opeq2 3597 | . . . . . . . . 9 ⊢ (𝑥 = (2nd ‘𝐴) → 〈𝐵, 𝑥〉 = 〈𝐵, (2nd ‘𝐴)〉) | |
8 | 7 | eqeq2d 2094 | . . . . . . . 8 ⊢ (𝑥 = (2nd ‘𝐴) → (𝐴 = 〈𝐵, 𝑥〉 ↔ 𝐴 = 〈𝐵, (2nd ‘𝐴)〉)) |
9 | 8 | spcegv 2697 | . . . . . . 7 ⊢ ((2nd ‘𝐴) ∈ V → (𝐴 = 〈𝐵, (2nd ‘𝐴)〉 → ∃𝑥 𝐴 = 〈𝐵, 𝑥〉)) |
10 | 6, 9 | syl 14 | . . . . . 6 ⊢ (𝐴 ∈ (V × V) → (𝐴 = 〈𝐵, (2nd ‘𝐴)〉 → ∃𝑥 𝐴 = 〈𝐵, 𝑥〉)) |
11 | 10 | adantr 270 | . . . . 5 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = 𝐵) → (𝐴 = 〈𝐵, (2nd ‘𝐴)〉 → ∃𝑥 𝐴 = 〈𝐵, 𝑥〉)) |
12 | 5, 11 | mpd 13 | . . . 4 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = 𝐵) → ∃𝑥 𝐴 = 〈𝐵, 𝑥〉) |
13 | 12 | ex 113 | . . 3 ⊢ (𝐴 ∈ (V × V) → ((1st ‘𝐴) = 𝐵 → ∃𝑥 𝐴 = 〈𝐵, 𝑥〉)) |
14 | eqop 5882 | . . . . 5 ⊢ (𝐴 ∈ (V × V) → (𝐴 = 〈𝐵, 𝑥〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝑥))) | |
15 | simpl 107 | . . . . 5 ⊢ (((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝑥) → (1st ‘𝐴) = 𝐵) | |
16 | 14, 15 | syl6bi 161 | . . . 4 ⊢ (𝐴 ∈ (V × V) → (𝐴 = 〈𝐵, 𝑥〉 → (1st ‘𝐴) = 𝐵)) |
17 | 16 | exlimdv 1742 | . . 3 ⊢ (𝐴 ∈ (V × V) → (∃𝑥 𝐴 = 〈𝐵, 𝑥〉 → (1st ‘𝐴) = 𝐵)) |
18 | 13, 17 | impbid 127 | . 2 ⊢ (𝐴 ∈ (V × V) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = 〈𝐵, 𝑥〉)) |
19 | 2, 18 | syl 14 | 1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = 〈𝐵, 𝑥〉)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1285 ∃wex 1422 ∈ wcel 1434 Vcvv 2612 〈cop 3425 × cxp 4399 ‘cfv 4969 1st c1st 5844 2nd c2nd 5845 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 ax-un 4224 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-sbc 2827 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-mpt 3867 df-id 4084 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-fo 4975 df-fv 4977 df-1st 5846 df-2nd 5847 |
This theorem is referenced by: releldm2 5890 |
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