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Theorem ot3rdg 7687
Description: Extract the third member of an ordered triple. (See ot1stg 7685 comment.) (Contributed by NM, 3-Apr-2015.)
Assertion
Ref Expression
ot3rdg (𝐶𝑉 → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)

Proof of Theorem ot3rdg
StepHypRef Expression
1 df-ot 4534 . . 3 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
21fveq2i 6648 . 2 (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
3 opex 5321 . . 3 𝐴, 𝐵⟩ ∈ V
4 op2ndg 7684 . . 3 ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶𝑉) → (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = 𝐶)
53, 4mpan 689 . 2 (𝐶𝑉 → (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = 𝐶)
62, 5syl5eq 2845 1 (𝐶𝑉 → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2111  Vcvv 3441  cop 4531  cotp 4533  cfv 6324  2nd c2nd 7670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-ot 4534  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fv 6332  df-2nd 7672
This theorem is referenced by:  oteqimp  7690  el2xptp0  7718  splval  14104  splcl  14105  ida2  17311  coa2  17321  mamufval  20992  msrval  32895  mapdhval  39017  hdmap1val  39091
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