Theorem ot3rdg 7982

Description: Extract the third member of an ordered triple. (See ot1stg 7980 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 4590	. . 3 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
2	1	fveq2i 6866	. 2 ⊢ (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
3		opex 5430	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
4		op2ndg 7979	. . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ 𝑉) → (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = 𝐶)
5	3, 4	mpan 700	. 2 ⊢ (𝐶 ∈ 𝑉 → (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = 𝐶)
6	2, 5	eqtrid 2808	1 ⊢ (𝐶 ∈ 𝑉 → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  Vcvv 3453  ⟨cop 4587  ⟨cotp 4589  ‘cfv 6517  2^nd c2nd 7965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-ot 4590  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-iota 6473  df-fun 6519  df-fv 6525  df-2nd 7967

This theorem is referenced by:  oteqimp  7985  el2xptp0  8013  sbcoteq1a  8028  xpord3lem  8124  splval  14761  splcl  14762  ida2  18075  coa2  18085  mamufval  22432  msrval  35852  mapdhval  42312  hdmap1val  42386