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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 7415 comment.) (Contributed by NM, 3-Apr-2015.) |
Ref | Expression |
---|---|
ot3rdg | ⊢ (𝐶 ∈ 𝑉 → (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ot 4377 | . . 3 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | 1 | fveq2i 6414 | . 2 ⊢ (2nd ‘〈𝐴, 𝐵, 𝐶〉) = (2nd ‘〈〈𝐴, 𝐵〉, 𝐶〉) |
3 | opex 5123 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
4 | op2ndg 7414 | . . 3 ⊢ ((〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ 𝑉) → (2nd ‘〈〈𝐴, 𝐵〉, 𝐶〉) = 𝐶) | |
5 | 3, 4 | mpan 682 | . 2 ⊢ (𝐶 ∈ 𝑉 → (2nd ‘〈〈𝐴, 𝐵〉, 𝐶〉) = 𝐶) |
6 | 2, 5 | syl5eq 2845 | 1 ⊢ (𝐶 ∈ 𝑉 → (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 Vcvv 3385 〈cop 4374 〈cotp 4376 ‘cfv 6101 2nd c2nd 7400 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-ot 4377 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-iota 6064 df-fun 6103 df-fv 6109 df-2nd 7402 |
This theorem is referenced by: oteqimp 7420 el2xptp0 7447 splvalpfxOLD 13823 splval 13824 splvalOLD 13825 splcl 13826 splclOLD 13827 ida2 17023 coa2 17033 mamufval 20516 msrval 31952 mapdhval 37745 hdmap1val 37819 |
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