Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  el2xptp0 Structured version   Visualization version   GIF version

Theorem el2xptp0 7745
 Description: A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
Assertion
Ref Expression
el2xptp0 ((𝑋𝑈𝑌𝑉𝑍𝑊) → ((𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍)) ↔ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩))

Proof of Theorem el2xptp0
StepHypRef Expression
1 xp1st 7730 . . . . . 6 (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (1st𝐴) ∈ (𝑈 × 𝑉))
21ad2antrl 727 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → (1st𝐴) ∈ (𝑈 × 𝑉))
3 3simpa 1145 . . . . . . 7 (((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍) → ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌))
43adantl 485 . . . . . 6 ((𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍)) → ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌))
54adantl 485 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌))
6 eqopi 7734 . . . . 5 (((1st𝐴) ∈ (𝑈 × 𝑉) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌)) → (1st𝐴) = ⟨𝑋, 𝑌⟩)
72, 5, 6syl2anc 587 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → (1st𝐴) = ⟨𝑋, 𝑌⟩)
8 simprr3 1220 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → (2nd𝐴) = 𝑍)
97, 8jca 515 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → ((1st𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2nd𝐴) = 𝑍))
10 df-ot 4534 . . . . . 6 𝑋, 𝑌, 𝑍⟩ = ⟨⟨𝑋, 𝑌⟩, 𝑍
1110eqeq2i 2771 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ 𝐴 = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩)
12 eqop 7740 . . . . 5 (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (𝐴 = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ↔ ((1st𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2nd𝐴) = 𝑍)))
1311, 12syl5bb 286 . . . 4 (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ ((1st𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2nd𝐴) = 𝑍)))
1413ad2antrl 727 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ ((1st𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2nd𝐴) = 𝑍)))
159, 14mpbird 260 . 2 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩)
16 opelxpi 5564 . . . . . . . 8 ((𝑋𝑈𝑌𝑉) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉))
17163adant3 1129 . . . . . . 7 ((𝑋𝑈𝑌𝑉𝑍𝑊) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉))
18 simp3 1135 . . . . . . 7 ((𝑋𝑈𝑌𝑉𝑍𝑊) → 𝑍𝑊)
1917, 18opelxpd 5565 . . . . . 6 ((𝑋𝑈𝑌𝑉𝑍𝑊) → ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
2010, 19eqeltrid 2856 . . . . 5 ((𝑋𝑈𝑌𝑉𝑍𝑊) → ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
2120adantr 484 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
22 eleq1 2839 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ↔ ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊)))
2322adantl 485 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ↔ ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊)))
2421, 23mpbird 260 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → 𝐴 ∈ ((𝑈 × 𝑉) × 𝑊))
25 2fveq3 6667 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (1st ‘(1st𝐴)) = (1st ‘(1st ‘⟨𝑋, 𝑌, 𝑍⟩)))
26 ot1stg 7712 . . . . 5 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (1st ‘(1st ‘⟨𝑋, 𝑌, 𝑍⟩)) = 𝑋)
2725, 26sylan9eqr 2815 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (1st ‘(1st𝐴)) = 𝑋)
28 2fveq3 6667 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (2nd ‘(1st𝐴)) = (2nd ‘(1st ‘⟨𝑋, 𝑌, 𝑍⟩)))
29 ot2ndg 7713 . . . . 5 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (2nd ‘(1st ‘⟨𝑋, 𝑌, 𝑍⟩)) = 𝑌)
3028, 29sylan9eqr 2815 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (2nd ‘(1st𝐴)) = 𝑌)
31 fveq2 6662 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (2nd𝐴) = (2nd ‘⟨𝑋, 𝑌, 𝑍⟩))
32 ot3rdg 7714 . . . . . 6 (𝑍𝑊 → (2nd ‘⟨𝑋, 𝑌, 𝑍⟩) = 𝑍)
33323ad2ant3 1132 . . . . 5 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (2nd ‘⟨𝑋, 𝑌, 𝑍⟩) = 𝑍)
3431, 33sylan9eqr 2815 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (2nd𝐴) = 𝑍)
3527, 30, 343jca 1125 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))
3624, 35jca 515 . 2 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍)))
3715, 36impbida 800 1 ((𝑋𝑈𝑌𝑉𝑍𝑊) → ((𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍)) ↔ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ⟨cop 4531  ⟨cotp 4533   × cxp 5525  ‘cfv 6339  1st c1st 7696  2nd c2nd 7697 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-ot 4534  df-uni 4802  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-iota 6298  df-fun 6341  df-fv 6347  df-1st 7698  df-2nd 7699 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator