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Theorem ot3rdgg 6045
Description: Extract the third member of an ordered triple. (See ot1stg 6043 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 3532 . . 3 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
21fveq2i 5417 . 2 (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
3 opexg 4145 . . . 4 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ ∈ V)
4 op2ndg 6042 . . . 4 ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶𝑋) → (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = 𝐶)
53, 4sylan 281 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐶𝑋) → (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = 𝐶)
653impa 1176 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = 𝐶)
72, 6syl5eq 2182 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 962   = wceq 1331  wcel 1480  Vcvv 2681  cop 3525  cotp 3526  cfv 5118  2nd c2nd 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-ot 3532  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fv 5126  df-2nd 6032
This theorem is referenced by: (None)
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