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Mirrors > Home > MPE Home > Th. List > ot2ndg | Structured version Visualization version GIF version |
Description: Extract the second member of an ordered triple. (See ot1stg 8027 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 4640 | . . . . . 6 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | 1 | fveq2i 6910 | . . . . 5 ⊢ (1st ‘〈𝐴, 𝐵, 𝐶〉) = (1st ‘〈〈𝐴, 𝐵〉, 𝐶〉) |
3 | opex 5475 | . . . . . 6 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
4 | op1stg 8025 | . . . . . 6 ⊢ ((〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ 𝑋) → (1st ‘〈〈𝐴, 𝐵〉, 𝐶〉) = 〈𝐴, 𝐵〉) | |
5 | 3, 4 | mpan 690 | . . . . 5 ⊢ (𝐶 ∈ 𝑋 → (1st ‘〈〈𝐴, 𝐵〉, 𝐶〉) = 〈𝐴, 𝐵〉) |
6 | 2, 5 | eqtrid 2787 | . . . 4 ⊢ (𝐶 ∈ 𝑋 → (1st ‘〈𝐴, 𝐵, 𝐶〉) = 〈𝐴, 𝐵〉) |
7 | 6 | fveq2d 6911 | . . 3 ⊢ (𝐶 ∈ 𝑋 → (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = (2nd ‘〈𝐴, 𝐵〉)) |
8 | op2ndg 8026 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) | |
9 | 7, 8 | sylan9eqr 2797 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵) |
10 | 9 | 3impa 1109 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 〈cotp 4639 ‘cfv 6563 1st c1st 8011 2nd c2nd 8012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-ot 4640 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-1st 8013 df-2nd 8014 |
This theorem is referenced by: oteqimp 8032 el2xptp0 8060 sbcoteq1a 8075 xpord3lem 8173 splval 14786 mamufval 22412 msrval 35523 mapdhval 41707 hdmap1val 41781 |
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