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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 7705 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 4578 | . . . . . 6 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | 1 | fveq2i 6675 | . . . . 5 ⊢ (1st ‘〈𝐴, 𝐵, 𝐶〉) = (1st ‘〈〈𝐴, 𝐵〉, 𝐶〉) |
3 | opex 5358 | . . . . . 6 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
4 | op1stg 7703 | . . . . . 6 ⊢ ((〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ 𝑋) → (1st ‘〈〈𝐴, 𝐵〉, 𝐶〉) = 〈𝐴, 𝐵〉) | |
5 | 3, 4 | mpan 688 | . . . . 5 ⊢ (𝐶 ∈ 𝑋 → (1st ‘〈〈𝐴, 𝐵〉, 𝐶〉) = 〈𝐴, 𝐵〉) |
6 | 2, 5 | syl5eq 2870 | . . . 4 ⊢ (𝐶 ∈ 𝑋 → (1st ‘〈𝐴, 𝐵, 𝐶〉) = 〈𝐴, 𝐵〉) |
7 | 6 | fveq2d 6676 | . . 3 ⊢ (𝐶 ∈ 𝑋 → (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = (2nd ‘〈𝐴, 𝐵〉)) |
8 | op2ndg 7704 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) | |
9 | 7, 8 | sylan9eqr 2880 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵) |
10 | 9 | 3impa 1106 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3496 〈cop 4575 〈cotp 4577 ‘cfv 6357 1st c1st 7689 2nd c2nd 7690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-ot 4578 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fv 6365 df-1st 7691 df-2nd 7692 |
This theorem is referenced by: oteqimp 7710 el2xptp0 7738 splval 14115 mamufval 20998 msrval 32787 mapdhval 38862 hdmap1val 38936 |
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