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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 7566 comment.) (Contributed by NM, 3-Apr-2015.) |
Ref | Expression |
---|---|
ot3rdg | ⊢ (𝐶 ∈ 𝑉 → (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ot 4487 | . . 3 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | 1 | fveq2i 6548 | . 2 ⊢ (2nd ‘〈𝐴, 𝐵, 𝐶〉) = (2nd ‘〈〈𝐴, 𝐵〉, 𝐶〉) |
3 | opex 5255 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
4 | op2ndg 7565 | . . 3 ⊢ ((〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ 𝑉) → (2nd ‘〈〈𝐴, 𝐵〉, 𝐶〉) = 𝐶) | |
5 | 3, 4 | mpan 686 | . 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 1525 ∈ wcel 2083 Vcvv 3440 〈cop 4484 〈cotp 4486 ‘cfv 6232 2nd c2nd 7551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-ot 4487 df-uni 4752 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-iota 6196 df-fun 6234 df-fv 6240 df-2nd 7553 |
This theorem is referenced by: oteqimp 7571 el2xptp0 7599 splval 13953 splcl 13954 ida2 17152 coa2 17162 mamufval 20682 msrval 32395 mapdhval 38412 hdmap1val 38486 |
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