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Theorem oteqimp 7172
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 7167 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
2 ot2ndg 7168 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)
3 ot3rdg 7169 . . . 4 (𝐶𝑍 → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
433ad2ant3 1082 . . 3 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
51, 2, 43jca 1240 . 2 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵 ∧ (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶))
6 fveq2 6178 . . . . 5 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (1st𝑇) = (1st ‘⟨𝐴, 𝐵, 𝐶⟩))
76fveq2d 6182 . . . 4 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (1st ‘(1st𝑇)) = (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)))
87eqeq1d 2622 . . 3 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((1st ‘(1st𝑇)) = 𝐴 ↔ (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴))
96fveq2d 6182 . . . 4 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (2nd ‘(1st𝑇)) = (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)))
109eqeq1d 2622 . . 3 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2nd ‘(1st𝑇)) = 𝐵 ↔ (2nd ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵))
11 fveq2 6178 . . . 4 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (2nd𝑇) = (2nd ‘⟨𝐴, 𝐵, 𝐶⟩))
1211eqeq1d 2622 . . 3 (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2nd𝑇) = 𝐶 ↔ (2nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶))
138, 10, 123anbi123d 1397 . 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 1036   = wceq 1481  wcel 1988  cotp 4176  cfv 5876  1st c1st 7151  2nd c2nd 7152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-ot 4177  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fv 5884  df-1st 7153  df-2nd 7154
This theorem is referenced by: (None)
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