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Mirrors > Home > ILE Home > Th. List > ot1stg | GIF version |
Description: Extract the first member of an ordered triple. (Due to infrequent usage, it isn't worthwhile at this point to define special extractors for triples, so we reuse the ordered pair extractors for ot1stg 6043, ot2ndg 6044, ot3rdgg 6045.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.) |
Ref | Expression |
---|---|
ot1stg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ot 3532 | . . . . 5 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | 1 | fveq2i 5417 | . . . 4 ⊢ (1st ‘〈𝐴, 𝐵, 𝐶〉) = (1st ‘〈〈𝐴, 𝐵〉, 𝐶〉) |
3 | opexg 4145 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ V) | |
4 | op1stg 6041 | . . . . . 6 ⊢ ((〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ 𝑋) → (1st ‘〈〈𝐴, 𝐵〉, 𝐶〉) = 〈𝐴, 𝐵〉) | |
5 | 3, 4 | sylan 281 | . . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → (1st ‘〈〈𝐴, 𝐵〉, 𝐶〉) = 〈𝐴, 𝐵〉) |
6 | 5 | 3impa 1176 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1st ‘〈〈𝐴, 𝐵〉, 𝐶〉) = 〈𝐴, 𝐵〉) |
7 | 2, 6 | syl5eq 2182 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1st ‘〈𝐴, 𝐵, 𝐶〉) = 〈𝐴, 𝐵〉) |
8 | 7 | fveq2d 5418 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = (1st ‘〈𝐴, 𝐵〉)) |
9 | op1stg 6041 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘〈𝐴, 𝐵〉) = 𝐴) | |
10 | 9 | 3adant3 1001 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1st ‘〈𝐴, 𝐵〉) = 𝐴) |
11 | 8, 10 | eqtrd 2170 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 Vcvv 2681 〈cop 3525 〈cotp 3526 ‘cfv 5118 1st c1st 6029 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-ot 3532 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fv 5126 df-1st 6031 |
This theorem is referenced by: (None) |
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