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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 4617 | . . 3 ⊢ (𝑉 × 𝑊) ⊆ (V × V) | |
2 | 1 | sseli 3063 | . 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V)) |
3 | eqid 2117 | . . . . . 6 ⊢ (2nd ‘𝐴) = (2nd ‘𝐴) | |
4 | eqopi 6038 | . . . . . 6 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = (2nd ‘𝐴))) → 𝐴 = 〈𝐵, (2nd ‘𝐴)〉) | |
5 | 3, 4 | mpanr2 434 | . . . . 5 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) = 𝐵) → 𝐴 = 〈𝐵, (2nd ‘𝐴)〉) |
6 | 2ndexg 6034 | . . . . . . 7 ⊢ (𝐴 ∈ (V × V) → (2nd ‘𝐴) ∈ V) | |
7 | opeq2 3676 | . . . . . . . . 9 ⊢ (𝑥 = (2nd ‘𝐴) → 〈𝐵, 𝑥〉 = 〈𝐵, (2nd ‘𝐴)〉) | |
8 | 7 | eqeq2d 2129 | . . . . . . . 8 ⊢ (𝑥 = (2nd ‘𝐴) → (𝐴 = 〈𝐵, 𝑥〉 ↔ 𝐴 = 〈𝐵, (2nd ‘𝐴)〉)) |
9 | 8 | spcegv 2748 | . . . . . . 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 274 | . . . . 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 114 | . . 3 ⊢ (𝐴 ∈ (V × V) → ((1st ‘𝐴) = 𝐵 → ∃𝑥 𝐴 = 〈𝐵, 𝑥〉)) |
14 | eqop 6043 | . . . . 5 ⊢ (𝐴 ∈ (V × V) → (𝐴 = 〈𝐵, 𝑥〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝑥))) | |
15 | simpl 108 | . . . . 5 ⊢ (((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝑥) → (1st ‘𝐴) = 𝐵) | |
16 | 14, 15 | syl6bi 162 | . . . 4 ⊢ (𝐴 ∈ (V × V) → (𝐴 = 〈𝐵, 𝑥〉 → (1st ‘𝐴) = 𝐵)) |
17 | 16 | exlimdv 1775 | . . 3 ⊢ (𝐴 ∈ (V × V) → (∃𝑥 𝐴 = 〈𝐵, 𝑥〉 → (1st ‘𝐴) = 𝐵)) |
18 | 13, 17 | impbid 128 | . 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 103 ↔ wb 104 = wceq 1316 ∃wex 1453 ∈ wcel 1465 Vcvv 2660 〈cop 3500 × cxp 4507 ‘cfv 5093 1st c1st 6004 2nd c2nd 6005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fo 5099 df-fv 5101 df-1st 6006 df-2nd 6007 |
This theorem is referenced by: releldm2 6051 |
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