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Theorem ot1stg 7989
Description: Extract the first member of an ordered triple. (Due to infrequent usage, it isn't worthwhile at this point to define special extractors for triples, so we reuse the ordered pair extractors for ot1stg 7989, ot2ndg 7990, ot3rdg 7991.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
Assertion
Ref Expression
ot1stg ((𝐴𝑉𝐵𝑊𝐶𝑋) → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)

Proof of Theorem ot1stg
StepHypRef Expression
1 df-ot 4638 . . . . . 6 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
21fveq2i 6895 . . . . 5 (1st ‘⟨𝐴, 𝐵, 𝐶⟩) = (1st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
3 opex 5465 . . . . . 6 𝐴, 𝐵⟩ ∈ V
4 op1stg 7987 . . . . . 6 ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶𝑋) → (1st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
53, 4mpan 689 . . . . 5 (𝐶𝑋 → (1st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
62, 5eqtrid 2785 . . . 4 (𝐶𝑋 → (1st ‘⟨𝐴, 𝐵, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
76fveq2d 6896 . . 3 (𝐶𝑋 → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = (1st ‘⟨𝐴, 𝐵⟩))
8 op1stg 7987 . . 3 ((𝐴𝑉𝐵𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
97, 8sylan9eqr 2795 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐶𝑋) → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
1093impa 1111 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3475  cop 4635  cotp 4637  cfv 6544  1st c1st 7973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-ot 4638  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-1st 7975
This theorem is referenced by:  oteqimp  7994  el2xptp0  8022  sbcoteq1a  8037  xpord3lem  8135  splval  14701  mamufval  21887  msrval  34529  elmsta  34539  mapdhval  40595  hdmap1val  40669
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