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Theorem ot1stg 7936
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 7936, ot2ndg 7937, ot3rdg 7938.) (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 4596 . . . . . 6 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
21fveq2i 6846 . . . . 5 (1st ‘⟨𝐴, 𝐵, 𝐶⟩) = (1st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
3 opex 5422 . . . . . 6 𝐴, 𝐵⟩ ∈ V
4 op1stg 7934 . . . . . 6 ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶𝑋) → (1st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
53, 4mpan 689 . . . . 5 (𝐶𝑋 → (1st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
62, 5eqtrid 2789 . . . 4 (𝐶𝑋 → (1st ‘⟨𝐴, 𝐵, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
76fveq2d 6847 . . 3 (𝐶𝑋 → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = (1st ‘⟨𝐴, 𝐵⟩))
8 op1stg 7934 . . 3 ((𝐴𝑉𝐵𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
97, 8sylan9eqr 2799 . 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 3446  cop 4593  cotp 4595  cfv 6497  1st c1st 7920
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 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-ot 4596  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-1st 7922
This theorem is referenced by:  oteqimp  7941  el2xptp0  7969  sbcoteq1a  7984  xpord3lem  8082  splval  14640  mamufval  21737  msrval  34135  elmsta  34145  mapdhval  40190  hdmap1val  40264
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