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Mirrors > Home > ILE Home > Th. List > ot2ndg | GIF version |
Description: Extract the second member of an ordered triple. (See ot1stg 6172 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.) |
Ref | Expression |
---|---|
ot2ndg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ot 3617 | . . . . 5 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | 1 | fveq2i 5534 | . . . 4 ⊢ (1st ‘〈𝐴, 𝐵, 𝐶〉) = (1st ‘〈〈𝐴, 𝐵〉, 𝐶〉) |
3 | opexg 4243 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ V) | |
4 | op1stg 6170 | . . . . . 6 ⊢ ((〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ 𝑋) → (1st ‘〈〈𝐴, 𝐵〉, 𝐶〉) = 〈𝐴, 𝐵〉) | |
5 | 3, 4 | sylan 283 | . . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → (1st ‘〈〈𝐴, 𝐵〉, 𝐶〉) = 〈𝐴, 𝐵〉) |
6 | 5 | 3impa 1196 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1st ‘〈〈𝐴, 𝐵〉, 𝐶〉) = 〈𝐴, 𝐵〉) |
7 | 2, 6 | eqtrid 2234 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1st ‘〈𝐴, 𝐵, 𝐶〉) = 〈𝐴, 𝐵〉) |
8 | 7 | fveq2d 5535 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = (2nd ‘〈𝐴, 𝐵〉)) |
9 | op2ndg 6171 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) | |
10 | 9 | 3adant3 1019 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) |
11 | 8, 10 | eqtrd 2222 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 Vcvv 2752 〈cop 3610 〈cotp 3611 ‘cfv 5232 1st c1st 6158 2nd c2nd 6159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-ot 3617 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-iota 5193 df-fun 5234 df-fv 5240 df-1st 6160 df-2nd 6161 |
This theorem is referenced by: (None) |
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