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Mirrors > Home > ILE Home > Th. List > op1stg | GIF version |
Description: Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.) |
Ref | Expression |
---|---|
op1stg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘〈𝐴, 𝐵〉) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 3675 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | fveq2d 5393 | . . 3 ⊢ (𝑥 = 𝐴 → (1st ‘〈𝑥, 𝑦〉) = (1st ‘〈𝐴, 𝑦〉)) |
3 | id 19 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
4 | 2, 3 | eqeq12d 2132 | . 2 ⊢ (𝑥 = 𝐴 → ((1st ‘〈𝑥, 𝑦〉) = 𝑥 ↔ (1st ‘〈𝐴, 𝑦〉) = 𝐴)) |
5 | opeq2 3676 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
6 | 5 | fveq2d 5393 | . . 3 ⊢ (𝑦 = 𝐵 → (1st ‘〈𝐴, 𝑦〉) = (1st ‘〈𝐴, 𝐵〉)) |
7 | 6 | eqeq1d 2126 | . 2 ⊢ (𝑦 = 𝐵 → ((1st ‘〈𝐴, 𝑦〉) = 𝐴 ↔ (1st ‘〈𝐴, 𝐵〉) = 𝐴)) |
8 | vex 2663 | . . 3 ⊢ 𝑥 ∈ V | |
9 | vex 2663 | . . 3 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | op1st 6012 | . 2 ⊢ (1st ‘〈𝑥, 𝑦〉) = 𝑥 |
11 | 4, 7, 10 | vtocl2g 2724 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘〈𝐴, 𝐵〉) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 〈cop 3500 ‘cfv 5093 1st c1st 6004 |
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-fv 5101 df-1st 6006 |
This theorem is referenced by: ot1stg 6018 ot2ndg 6019 1stconst 6086 algrflemg 6095 mpoxopn0yelv 6104 mpoxopoveq 6105 xpmapenlem 6711 1stinl 6927 1stinr 6929 mulpipq 7148 suplocexprlemlub 7500 aprcl 8376 frecuzrdgg 10157 qredeu 11705 qnumdenbi 11797 upxp 12368 uptx 12370 txmetcnp 12614 |
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