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Theorem oteqimp 7338
 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 7333 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
2 ot2ndg 7334 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)
3 ot3rdg 7335 . . . 4 (𝐶𝑍 → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
433ad2ant3 1129 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
51, 2, 43jca 1122 . 2 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵 ∧ (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶))
6 fveq2 6333 . . . . 5 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (1st𝑇) = (1st ‘⟨𝐴, 𝐵, 𝐶⟩))
76fveq2d 6337 . . . 4 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (1st ‘(1st𝑇)) = (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)))
87eqeq1d 2773 . . 3 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((1st ‘(1st𝑇)) = 𝐴 ↔ (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴))
96fveq2d 6337 . . . 4 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (2nd ‘(1st𝑇)) = (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)))
109eqeq1d 2773 . . 3 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2nd ‘(1st𝑇)) = 𝐵 ↔ (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵))
11 fveq2 6333 . . . 4 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (2nd𝑇) = (2nd ‘⟨𝐴, 𝐵, 𝐶⟩))
1211eqeq1d 2773 . . 3 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2nd𝑇) = 𝐶 ↔ (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶))
138, 10, 123anbi123d 1547 . 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 236 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 1071   = wceq 1631   ∈ wcel 2145  ⟨cotp 4325  ‘cfv 6030  1st c1st 7317  2nd c2nd 7318 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-ot 4326  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fv 6038  df-1st 7319  df-2nd 7320 This theorem is referenced by: (None)
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