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| Description: Extract the second member of an ordered triple. (See ot1stg 8028 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 4635 | . . . . . 6 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
| 2 | 1 | fveq2i 6909 | . . . . 5 ⊢ (1st ‘〈𝐴, 𝐵, 𝐶〉) = (1st ‘〈〈𝐴, 𝐵〉, 𝐶〉) | 
| 3 | opex 5469 | . . . . . 6 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
| 4 | op1stg 8026 | . . . . . 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 2789 | . . . 4 ⊢ (𝐶 ∈ 𝑋 → (1st ‘〈𝐴, 𝐵, 𝐶〉) = 〈𝐴, 𝐵〉) | 
| 7 | 6 | fveq2d 6910 | . . 3 ⊢ (𝐶 ∈ 𝑋 → (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = (2nd ‘〈𝐴, 𝐵〉)) | 
| 8 | op2ndg 8027 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) | |
| 9 | 7, 8 | sylan9eqr 2799 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵) | 
| 10 | 9 | 3impa 1110 | 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 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 〈cotp 4634 ‘cfv 6561 1st c1st 8012 2nd c2nd 8013 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-ot 4635 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-1st 8014 df-2nd 8015 | 
| This theorem is referenced by: oteqimp 8033 el2xptp0 8061 sbcoteq1a 8076 xpord3lem 8174 splval 14789 mamufval 22396 msrval 35543 mapdhval 41726 hdmap1val 41800 | 
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