Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  oteqimp Structured version   Visualization version   GIF version

Theorem oteqimp 7078
 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 7073 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
2 ot2ndg 7074 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)
3 ot3rdg 7075 . . . 4 (𝐶𝑍 → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
433ad2ant3 1077 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
51, 2, 43jca 1235 . 2 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵 ∧ (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶))
6 fveq2 6103 . . . . 5 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (1st𝑇) = (1st ‘⟨𝐴, 𝐵, 𝐶⟩))
76fveq2d 6107 . . . 4 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (1st ‘(1st𝑇)) = (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)))
87eqeq1d 2612 . . 3 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((1st ‘(1st𝑇)) = 𝐴 ↔ (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴))
96fveq2d 6107 . . . 4 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (2nd ‘(1st𝑇)) = (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)))
109eqeq1d 2612 . . 3 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2nd ‘(1st𝑇)) = 𝐵 ↔ (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵))
11 fveq2 6103 . . . 4 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (2nd𝑇) = (2nd ‘⟨𝐴, 𝐵, 𝐶⟩))
1211eqeq1d 2612 . . 3 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2nd𝑇) = 𝐶 ↔ (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶))
138, 10, 123anbi123d 1391 . 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)))
145, 13syl5ibr 235 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 1031   = wceq 1475   ∈ wcel 1977  ⟨cotp 4133  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-1st 7059  df-2nd 7060 This theorem is referenced by:  el2wlkonot  26396  el2spthonot  26397  el2spthonot0  26398
 Copyright terms: Public domain W3C validator