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Theorem ot3rdg 8030
Description: Extract the third member of an ordered triple. (See ot1stg 8028 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 4635 . . 3 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
21fveq2i 6909 . 2 (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
3 opex 5469 . . 3 𝐴, 𝐵⟩ ∈ V
4 op2ndg 8027 . . 3 ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶𝑉) → (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = 𝐶)
53, 4mpan 690 . 2 (𝐶𝑉 → (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = 𝐶)
62, 5eqtrid 2789 1 (𝐶𝑉 → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632  cotp 4634  cfv 6561  2nd c2nd 8013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-2nd 8015
This theorem is referenced by:  oteqimp  8033  el2xptp0  8061  sbcoteq1a  8076  xpord3lem  8174  splval  14789  splcl  14790  ida2  18104  coa2  18114  mamufval  22396  msrval  35543  mapdhval  41726  hdmap1val  41800
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