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Mirrors > Home > MPE Home > Th. List > ot3rdg | Structured version Visualization version GIF version |
Description: Extract the third member of an ordered triple. (See ot1stg 7753 comment.) (Contributed by NM, 3-Apr-2015.) |
Ref | Expression |
---|---|
ot3rdg | ⊢ (𝐶 ∈ 𝑉 → (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ot 4536 | . . 3 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | 1 | fveq2i 6698 | . 2 ⊢ (2nd ‘〈𝐴, 𝐵, 𝐶〉) = (2nd ‘〈〈𝐴, 𝐵〉, 𝐶〉) |
3 | opex 5333 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
4 | op2ndg 7752 | . . 3 ⊢ ((〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ 𝑉) → (2nd ‘〈〈𝐴, 𝐵〉, 𝐶〉) = 𝐶) | |
5 | 3, 4 | mpan 690 | . 2 ⊢ (𝐶 ∈ 𝑉 → (2nd ‘〈〈𝐴, 𝐵〉, 𝐶〉) = 𝐶) |
6 | 2, 5 | syl5eq 2783 | 1 ⊢ (𝐶 ∈ 𝑉 → (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 Vcvv 3398 〈cop 4533 〈cotp 4535 ‘cfv 6358 2nd c2nd 7738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-ot 4536 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fv 6366 df-2nd 7740 |
This theorem is referenced by: oteqimp 7758 el2xptp0 7786 splval 14281 splcl 14282 ida2 17519 coa2 17529 mamufval 21238 msrval 33167 mapdhval 39424 hdmap1val 39498 |
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