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Theorem ot2ndg 8028
Description: Extract the second member of an ordered triple. (See ot1stg 8027 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
Assertion
Ref Expression
ot2ndg ((𝐴𝑉𝐵𝑊𝐶𝑋) → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)

Proof of Theorem ot2ndg
StepHypRef Expression
1 df-ot 4640 . . . . . 6 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
21fveq2i 6910 . . . . 5 (1st ‘⟨𝐴, 𝐵, 𝐶⟩) = (1st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
3 opex 5475 . . . . . 6 𝐴, 𝐵⟩ ∈ V
4 op1stg 8025 . . . . . 6 ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶𝑋) → (1st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
53, 4mpan 690 . . . . 5 (𝐶𝑋 → (1st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
62, 5eqtrid 2787 . . . 4 (𝐶𝑋 → (1st ‘⟨𝐴, 𝐵, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
76fveq2d 6911 . . 3 (𝐶𝑋 → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = (2nd ‘⟨𝐴, 𝐵⟩))
8 op2ndg 8026 . . 3 ((𝐴𝑉𝐵𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
97, 8sylan9eqr 2797 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐶𝑋) → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)
1093impa 1109 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637  cotp 4639  cfv 6563  1st c1st 8011  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-ot 4640  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-2nd 8014
This theorem is referenced by:  oteqimp  8032  el2xptp0  8060  sbcoteq1a  8075  xpord3lem  8173  splval  14786  mamufval  22412  msrval  35523  mapdhval  41707  hdmap1val  41781
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