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Theorem el2xptp0 8022
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 8007 . . . . . 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 1149 . . . . . . 7 (((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍) → ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌))
43adantl 483 . . . . . 6 ((𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍)) → ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌))
54adantl 483 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌))
6 eqopi 8011 . . . . 5 (((1st𝐴) ∈ (𝑈 × 𝑉) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌)) → (1st𝐴) = ⟨𝑋, 𝑌⟩)
72, 5, 6syl2anc 585 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → (1st𝐴) = ⟨𝑋, 𝑌⟩)
8 simprr3 1224 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → (2nd𝐴) = 𝑍)
97, 8jca 513 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → ((1st𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2nd𝐴) = 𝑍))
10 df-ot 4638 . . . . . 6 𝑋, 𝑌, 𝑍⟩ = ⟨⟨𝑋, 𝑌⟩, 𝑍
1110eqeq2i 2746 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ 𝐴 = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩)
12 eqop 8017 . . . . 5 (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (𝐴 = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ↔ ((1st𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2nd𝐴) = 𝑍)))
1311, 12bitrid 283 . . . 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 257 . 2 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩)
16 opelxpi 5714 . . . . . . . 8 ((𝑋𝑈𝑌𝑉) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉))
17163adant3 1133 . . . . . . 7 ((𝑋𝑈𝑌𝑉𝑍𝑊) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉))
18 simp3 1139 . . . . . . 7 ((𝑋𝑈𝑌𝑉𝑍𝑊) → 𝑍𝑊)
1917, 18opelxpd 5716 . . . . . 6 ((𝑋𝑈𝑌𝑉𝑍𝑊) → ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
2010, 19eqeltrid 2838 . . . . 5 ((𝑋𝑈𝑌𝑉𝑍𝑊) → ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
2120adantr 482 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
22 eleq1 2822 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ↔ ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊)))
2322adantl 483 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ↔ ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊)))
2421, 23mpbird 257 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → 𝐴 ∈ ((𝑈 × 𝑉) × 𝑊))
25 2fveq3 6897 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (1st ‘(1st𝐴)) = (1st ‘(1st ‘⟨𝑋, 𝑌, 𝑍⟩)))
26 ot1stg 7989 . . . . 5 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (1st ‘(1st ‘⟨𝑋, 𝑌, 𝑍⟩)) = 𝑋)
2725, 26sylan9eqr 2795 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (1st ‘(1st𝐴)) = 𝑋)
28 2fveq3 6897 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (2nd ‘(1st𝐴)) = (2nd ‘(1st ‘⟨𝑋, 𝑌, 𝑍⟩)))
29 ot2ndg 7990 . . . . 5 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (2nd ‘(1st ‘⟨𝑋, 𝑌, 𝑍⟩)) = 𝑌)
3028, 29sylan9eqr 2795 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (2nd ‘(1st𝐴)) = 𝑌)
31 fveq2 6892 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (2nd𝐴) = (2nd ‘⟨𝑋, 𝑌, 𝑍⟩))
32 ot3rdg 7991 . . . . . 6 (𝑍𝑊 → (2nd ‘⟨𝑋, 𝑌, 𝑍⟩) = 𝑍)
33323ad2ant3 1136 . . . . 5 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (2nd ‘⟨𝑋, 𝑌, 𝑍⟩) = 𝑍)
3431, 33sylan9eqr 2795 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (2nd𝐴) = 𝑍)
3527, 30, 343jca 1129 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))
3624, 35jca 513 . 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  cop 4635  cotp 4637   × cxp 5675  cfv 6544  1st c1st 7973  2nd c2nd 7974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-ot 4638  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-1st 7975  df-2nd 7976
This theorem is referenced by: (None)
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