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Theorem ot3rdgg 5925
Description: Extract the third member of an ordered triple. (See ot1stg 5923 comment.) (Contributed by NM, 3-Apr-2015.)
Assertion
Ref Expression
ot3rdgg ((𝐴𝑉𝐵𝑊𝐶𝑋) → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)

Proof of Theorem ot3rdgg
StepHypRef Expression
1 df-ot 3456 . . 3 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
21fveq2i 5308 . 2 (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
3 opexg 4055 . . . 4 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ ∈ V)
4 op2ndg 5922 . . . 4 ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶𝑋) → (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = 𝐶)
53, 4sylan 277 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐶𝑋) → (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = 𝐶)
653impa 1138 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = 𝐶)
72, 6syl5eq 2132 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 924   = wceq 1289  wcel 1438  Vcvv 2619  cop 3449  cotp 3450  cfv 5015  2nd c2nd 5910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-ot 3456  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fv 5023  df-2nd 5912
This theorem is referenced by: (None)
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