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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 7702 comment.) (Contributed by NM, 3-Apr-2015.) |
Ref | Expression |
---|---|
ot3rdg | ⊢ (𝐶 ∈ 𝑉 → (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ot 4575 | . . 3 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | 1 | fveq2i 6672 | . 2 ⊢ (2nd ‘〈𝐴, 𝐵, 𝐶〉) = (2nd ‘〈〈𝐴, 𝐵〉, 𝐶〉) |
3 | opex 5355 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
4 | op2ndg 7701 | . . 3 ⊢ ((〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ 𝑉) → (2nd ‘〈〈𝐴, 𝐵〉, 𝐶〉) = 𝐶) | |
5 | 3, 4 | mpan 688 | . 2 ⊢ (𝐶 ∈ 𝑉 → (2nd ‘〈〈𝐴, 𝐵〉, 𝐶〉) = 𝐶) |
6 | 2, 5 | syl5eq 2868 | 1 ⊢ (𝐶 ∈ 𝑉 → (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 〈cop 4572 〈cotp 4574 ‘cfv 6354 2nd c2nd 7687 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-ot 4575 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-iota 6313 df-fun 6356 df-fv 6362 df-2nd 7689 |
This theorem is referenced by: oteqimp 7707 el2xptp0 7735 splval 14112 splcl 14113 ida2 17318 coa2 17328 mamufval 20995 msrval 32785 mapdhval 38859 hdmap1val 38933 |
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