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Theorem oteqimp 7990
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 7985 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
2 ot2ndg 7986 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)
3 ot3rdg 7987 . . . 4 (𝐶𝑍 → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
433ad2ant3 1132 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
51, 2, 43jca 1125 . 2 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵 ∧ (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶))
6 2fveq3 6889 . . . 4 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (1st ‘(1st𝑇)) = (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)))
76eqeq1d 2728 . . 3 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((1st ‘(1st𝑇)) = 𝐴 ↔ (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴))
8 2fveq3 6889 . . . 4 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (2nd ‘(1st𝑇)) = (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)))
98eqeq1d 2728 . . 3 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2nd ‘(1st𝑇)) = 𝐵 ↔ (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵))
10 fveqeq2 6893 . . 3 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2nd𝑇) = 𝐶 ↔ (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶))
117, 9, 103anbi123d 1432 . 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 245 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 1084   = wceq 1533  wcel 2098  cotp 4631  cfv 6536  1st c1st 7969  2nd c2nd 7970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-ot 4632  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fv 6544  df-1st 7971  df-2nd 7972
This theorem is referenced by: (None)
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