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Theorem ot2ndg 6173
Description: Extract the second member of an ordered triple. (See ot1stg 6172 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 3617 . . . . 5 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
21fveq2i 5534 . . . 4 (1st ‘⟨𝐴, 𝐵, 𝐶⟩) = (1st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
3 opexg 4243 . . . . . 6 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ ∈ V)
4 op1stg 6170 . . . . . 6 ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶𝑋) → (1st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
53, 4sylan 283 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ 𝐶𝑋) → (1st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
653impa 1196 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (1st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
72, 6eqtrid 2234 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (1st ‘⟨𝐴, 𝐵, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
87fveq2d 5535 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = (2nd ‘⟨𝐴, 𝐵⟩))
9 op2ndg 6171 . . 3 ((𝐴𝑉𝐵𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
1093adant3 1019 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
118, 10eqtrd 2222 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980   = wceq 1364  wcel 2160  Vcvv 2752  cop 3610  cotp 3611  cfv 5232  1st c1st 6158  2nd c2nd 6159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-ot 3617  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-iota 5193  df-fun 5234  df-fv 5240  df-1st 6160  df-2nd 6161
This theorem is referenced by: (None)
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