Theorem ot3rdg 7985

Description: Extract the third member of an ordered triple. (See ot1stg 7983 comment.) (Contributed by NM, 3-Apr-2015.)

Assertion

Ref	Expression
ot3rdg	⊢ (𝐶 ∈ 𝑉 → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)

Proof of Theorem ot3rdg

Step	Hyp	Ref	Expression
1		df-ot 4630	. . 3 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
2	1	fveq2i 6885	. 2 ⊢ (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
3		opex 5455	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
4		op2ndg 7982	. . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ 𝑉) → (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = 𝐶)
5	3, 4	mpan 687	. 2 ⊢ (𝐶 ∈ 𝑉 → (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = 𝐶)
6	2, 5	eqtrid 2776	1 ⊢ (𝐶 ∈ 𝑉 → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3466  ⟨cop 4627  ⟨cotp 4629  ‘cfv 6534  2^nd c2nd 7968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-ot 4630  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fv 6542  df-2nd 7970

This theorem is referenced by:  oteqimp  7988  el2xptp0  8016  sbcoteq1a  8031  xpord3lem  8130  splval  14703  splcl  14704  ida2  18017  coa2  18027  mamufval  22231  msrval  35047  mapdhval  41099  hdmap1val  41173