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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 3765 | . . . 4 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑦〉 = 〈𝐴, 𝑦〉) | |
2 | 1 | fveq2d 5500 | . . 3 ⊢ (𝑥 = 𝐴 → (1st ‘〈𝑥, 𝑦〉) = (1st ‘〈𝐴, 𝑦〉)) |
3 | id 19 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
4 | 2, 3 | eqeq12d 2185 | . 2 ⊢ (𝑥 = 𝐴 → ((1st ‘〈𝑥, 𝑦〉) = 𝑥 ↔ (1st ‘〈𝐴, 𝑦〉) = 𝐴)) |
5 | opeq2 3766 | . . . 4 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
6 | 5 | fveq2d 5500 | . . 3 ⊢ (𝑦 = 𝐵 → (1st ‘〈𝐴, 𝑦〉) = (1st ‘〈𝐴, 𝐵〉)) |
7 | 6 | eqeq1d 2179 | . 2 ⊢ (𝑦 = 𝐵 → ((1st ‘〈𝐴, 𝑦〉) = 𝐴 ↔ (1st ‘〈𝐴, 𝐵〉) = 𝐴)) |
8 | vex 2733 | . . 3 ⊢ 𝑥 ∈ V | |
9 | vex 2733 | . . 3 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | op1st 6125 | . 2 ⊢ (1st ‘〈𝑥, 𝑦〉) = 𝑥 |
11 | 4, 7, 10 | vtocl2g 2794 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘〈𝐴, 𝐵〉) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 〈cop 3586 ‘cfv 5198 1st c1st 6117 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-iota 5160 df-fun 5200 df-fv 5206 df-1st 6119 |
This theorem is referenced by: ot1stg 6131 ot2ndg 6132 1stconst 6200 algrflemg 6209 mpoxopn0yelv 6218 mpoxopoveq 6219 xpmapenlem 6827 1stinl 7051 1stinr 7053 mulpipq 7334 suplocexprlemlub 7686 aprcl 8565 frecuzrdgg 10372 qredeu 12051 qnumdenbi 12146 upxp 13066 uptx 13068 txmetcnp 13312 |
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