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Theorem el2xptp0 7732
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 7717 . . . . . 6 (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (1st𝐴) ∈ (𝑈 × 𝑉))
21ad2antrl 724 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → (1st𝐴) ∈ (𝑈 × 𝑉))
3 3simpa 1142 . . . . . . 7 (((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍) → ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌))
43adantl 482 . . . . . 6 ((𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍)) → ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌))
54adantl 482 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌))
6 eqopi 7721 . . . . 5 (((1st𝐴) ∈ (𝑈 × 𝑉) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌)) → (1st𝐴) = ⟨𝑋, 𝑌⟩)
72, 5, 6syl2anc 584 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → (1st𝐴) = ⟨𝑋, 𝑌⟩)
8 simprr3 1217 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → (2nd𝐴) = 𝑍)
97, 8jca 512 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → ((1st𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2nd𝐴) = 𝑍))
10 df-ot 4573 . . . . . 6 𝑋, 𝑌, 𝑍⟩ = ⟨⟨𝑋, 𝑌⟩, 𝑍
1110eqeq2i 2839 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ 𝐴 = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩)
12 eqop 7727 . . . . 5 (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (𝐴 = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ↔ ((1st𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2nd𝐴) = 𝑍)))
1311, 12syl5bb 284 . . . 4 (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ ((1st𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2nd𝐴) = 𝑍)))
1413ad2antrl 724 . . 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 258 . 2 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩)
16 opelxpi 5591 . . . . . . . 8 ((𝑋𝑈𝑌𝑉) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉))
17163adant3 1126 . . . . . . 7 ((𝑋𝑈𝑌𝑉𝑍𝑊) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉))
18 simp3 1132 . . . . . . 7 ((𝑋𝑈𝑌𝑉𝑍𝑊) → 𝑍𝑊)
1917, 18opelxpd 5592 . . . . . 6 ((𝑋𝑈𝑌𝑉𝑍𝑊) → ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
2010, 19eqeltrid 2922 . . . . 5 ((𝑋𝑈𝑌𝑉𝑍𝑊) → ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
2120adantr 481 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
22 eleq1 2905 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ↔ ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊)))
2322adantl 482 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ↔ ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊)))
2421, 23mpbird 258 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → 𝐴 ∈ ((𝑈 × 𝑉) × 𝑊))
25 2fveq3 6674 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (1st ‘(1st𝐴)) = (1st ‘(1st ‘⟨𝑋, 𝑌, 𝑍⟩)))
26 ot1stg 7699 . . . . 5 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (1st ‘(1st ‘⟨𝑋, 𝑌, 𝑍⟩)) = 𝑋)
2725, 26sylan9eqr 2883 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (1st ‘(1st𝐴)) = 𝑋)
28 2fveq3 6674 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (2nd ‘(1st𝐴)) = (2nd ‘(1st ‘⟨𝑋, 𝑌, 𝑍⟩)))
29 ot2ndg 7700 . . . . 5 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (2nd ‘(1st ‘⟨𝑋, 𝑌, 𝑍⟩)) = 𝑌)
3028, 29sylan9eqr 2883 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (2nd ‘(1st𝐴)) = 𝑌)
31 fveq2 6669 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (2nd𝐴) = (2nd ‘⟨𝑋, 𝑌, 𝑍⟩))
32 ot3rdg 7701 . . . . . 6 (𝑍𝑊 → (2nd ‘⟨𝑋, 𝑌, 𝑍⟩) = 𝑍)
33323ad2ant3 1129 . . . . 5 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (2nd ‘⟨𝑋, 𝑌, 𝑍⟩) = 𝑍)
3431, 33sylan9eqr 2883 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (2nd𝐴) = 𝑍)
3527, 30, 343jca 1122 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))
3624, 35jca 512 . 2 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍)))
3715, 36impbida 797 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  cop 4570  cotp 4572   × cxp 5552  cfv 6354  1st c1st 7683  2nd c2nd 7684
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-ot 4573  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fv 6362  df-1st 7685  df-2nd 7686
This theorem is referenced by: (None)
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