Theorem ot3rdg 7987

Description: Extract the third member of an ordered triple. (See ot1stg 7985 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 4601	. . 3 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
2	1	fveq2i 6864	. 2 ⊢ (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
3		opex 5427	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
4		op2ndg 7984	. . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ 𝑉) → (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = 𝐶)
5	3, 4	mpan 690	. 2 ⊢ (𝐶 ∈ 𝑉 → (2nd ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = 𝐶)
6	2, 5	eqtrid 2777	1 ⊢ (𝐶 ∈ 𝑉 → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ⟨cop 4598  ⟨cotp 4600  ‘cfv 6514  2^nd c2nd 7970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-ot 4601  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fv 6522  df-2nd 7972

This theorem is referenced by:  oteqimp  7990  el2xptp0  8018  sbcoteq1a  8033  xpord3lem  8131  splval  14723  splcl  14724  ida2  18028  coa2  18038  mamufval  22286  msrval  35532  mapdhval  41725  hdmap1val  41799