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Theorem ot3rdgg 5723
Description: Extract the third member of an ordered triple. (See ot1stg 5721 comment.) (Contributed by NM, 3-Apr-2015.)
Assertion
Ref Expression
ot3rdgg ((A 𝑉 B 𝑊 𝐶 𝑋) → (2nd ‘⟨A, B, 𝐶⟩) = 𝐶)

Proof of Theorem ot3rdgg
StepHypRef Expression
1 df-ot 3377 . . 3 A, B, 𝐶⟩ = ⟨⟨A, B⟩, 𝐶
21fveq2i 5124 . 2 (2nd ‘⟨A, B, 𝐶⟩) = (2nd ‘⟨⟨A, B⟩, 𝐶⟩)
3 opexg 3955 . . . 4 ((A 𝑉 B 𝑊) → ⟨A, B V)
4 op2ndg 5720 . . . 4 ((⟨A, B V 𝐶 𝑋) → (2nd ‘⟨⟨A, B⟩, 𝐶⟩) = 𝐶)
53, 4sylan 267 . . 3 (((A 𝑉 B 𝑊) 𝐶 𝑋) → (2nd ‘⟨⟨A, B⟩, 𝐶⟩) = 𝐶)
653impa 1098 . 2 ((A 𝑉 B 𝑊 𝐶 𝑋) → (2nd ‘⟨⟨A, B⟩, 𝐶⟩) = 𝐶)
72, 6syl5eq 2081 1 ((A 𝑉 B 𝑊 𝐶 𝑋) → (2nd ‘⟨A, B, 𝐶⟩) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242   wcel 1390  Vcvv 2551  cop 3370  cotp 3371  cfv 4845  2nd c2nd 5708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-ot 3377  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fv 4853  df-2nd 5710
This theorem is referenced by: (None)
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