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Theorem oteqimp 8032
Description: The components of an ordered triple. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
Assertion
Ref Expression
oteqimp (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = 𝐵 ∧ (2nd𝑇) = 𝐶)))

Proof of Theorem oteqimp
StepHypRef Expression
1 ot1stg 8027 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
2 ot2ndg 8028 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)
3 ot3rdg 8029 . . . 4 (𝐶𝑍 → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
433ad2ant3 1134 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
51, 2, 43jca 1127 . 2 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵 ∧ (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶))
6 2fveq3 6912 . . . 4 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (1st ‘(1st𝑇)) = (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)))
76eqeq1d 2737 . . 3 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((1st ‘(1st𝑇)) = 𝐴 ↔ (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴))
8 2fveq3 6912 . . . 4 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (2nd ‘(1st𝑇)) = (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)))
98eqeq1d 2737 . . 3 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2nd ‘(1st𝑇)) = 𝐵 ↔ (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵))
10 fveqeq2 6916 . . 3 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2nd𝑇) = 𝐶 ↔ (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶))
117, 9, 103anbi123d 1435 . 2 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = 𝐵 ∧ (2nd𝑇) = 𝐶) ↔ ((1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵 ∧ (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)))
125, 11imbitrrid 246 1 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = 𝐵 ∧ (2nd𝑇) = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  cotp 4639  cfv 6563  1st c1st 8011  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-ot 4640  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-2nd 8014
This theorem is referenced by: (None)
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