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Theorem ot3rdg 7938
Description: Extract the third member of an ordered triple. (See ot1stg 7936 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 4596 . . 3 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
21fveq2i 6846 . 2 (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
3 opex 5422 . . 3 𝐴, 𝐵⟩ ∈ V
4 op2ndg 7935 . . 3 ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶𝑉) → (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = 𝐶)
53, 4mpan 689 . 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 1542  wcel 2107  Vcvv 3446  cop 4593  cotp 4595  cfv 6497  2nd c2nd 7921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-ot 4596  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-2nd 7923
This theorem is referenced by:  oteqimp  7941  el2xptp0  7969  sbcoteq1a  7984  xpord3lem  8082  splval  14640  splcl  14641  ida2  17946  coa2  17956  mamufval  21737  msrval  34135  mapdhval  40190  hdmap1val  40264
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