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Theorem el2xptp0 7197
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 7183 . . . . . 6 (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (1st𝐴) ∈ (𝑈 × 𝑉))
21ad2antrl 763 . . . . 5 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → (1st𝐴) ∈ (𝑈 × 𝑉))
3 3simpa 1056 . . . . . . 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 7187 . . . . 5 (((1st𝐴) ∈ (𝑈 × 𝑉) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌)) → (1st𝐴) = ⟨𝑋, 𝑌⟩)
72, 5, 6syl2anc 692 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → (1st𝐴) = ⟨𝑋, 𝑌⟩)
8 simprr3 1109 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → (2nd𝐴) = 𝑍)
97, 8jca 554 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → ((1st𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2nd𝐴) = 𝑍))
10 df-ot 4177 . . . . . 6 𝑋, 𝑌, 𝑍⟩ = ⟨⟨𝑋, 𝑌⟩, 𝑍
1110eqeq2i 2632 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ 𝐴 = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩)
12 eqop 7193 . . . . 5 (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (𝐴 = ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ↔ ((1st𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2nd𝐴) = 𝑍)))
1311, 12syl5bb 272 . . . 4 (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) → (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ ↔ ((1st𝐴) = ⟨𝑋, 𝑌⟩ ∧ (2nd𝐴) = 𝑍)))
1413ad2antrl 763 . . 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 247 . 2 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))) → 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩)
16 opelxpi 5138 . . . . . . . 8 ((𝑋𝑈𝑌𝑉) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉))
17163adant3 1079 . . . . . . 7 ((𝑋𝑈𝑌𝑉𝑍𝑊) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉))
18 simp3 1061 . . . . . . 7 ((𝑋𝑈𝑌𝑉𝑍𝑊) → 𝑍𝑊)
19 opelxp 5136 . . . . . . 7 (⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊) ↔ (⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑉) ∧ 𝑍𝑊))
2017, 18, 19sylanbrc 697 . . . . . 6 ((𝑋𝑈𝑌𝑉𝑍𝑊) → ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
2110, 20syl5eqel 2703 . . . . 5 ((𝑋𝑈𝑌𝑉𝑍𝑊) → ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
2221adantr 481 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊))
23 eleq1 2687 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ↔ ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊)))
2423adantl 482 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ↔ ⟨𝑋, 𝑌, 𝑍⟩ ∈ ((𝑈 × 𝑉) × 𝑊)))
2522, 24mpbird 247 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → 𝐴 ∈ ((𝑈 × 𝑉) × 𝑊))
26 fveq2 6178 . . . . . 6 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (1st𝐴) = (1st ‘⟨𝑋, 𝑌, 𝑍⟩))
2726fveq2d 6182 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (1st ‘(1st𝐴)) = (1st ‘(1st ‘⟨𝑋, 𝑌, 𝑍⟩)))
28 ot1stg 7167 . . . . 5 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (1st ‘(1st ‘⟨𝑋, 𝑌, 𝑍⟩)) = 𝑋)
2927, 28sylan9eqr 2676 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (1st ‘(1st𝐴)) = 𝑋)
3026fveq2d 6182 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (2nd ‘(1st𝐴)) = (2nd ‘(1st ‘⟨𝑋, 𝑌, 𝑍⟩)))
31 ot2ndg 7168 . . . . 5 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (2nd ‘(1st ‘⟨𝑋, 𝑌, 𝑍⟩)) = 𝑌)
3230, 31sylan9eqr 2676 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (2nd ‘(1st𝐴)) = 𝑌)
33 fveq2 6178 . . . . 5 (𝐴 = ⟨𝑋, 𝑌, 𝑍⟩ → (2nd𝐴) = (2nd ‘⟨𝑋, 𝑌, 𝑍⟩))
34 ot3rdg 7169 . . . . . 6 (𝑍𝑊 → (2nd ‘⟨𝑋, 𝑌, 𝑍⟩) = 𝑍)
35343ad2ant3 1082 . . . . 5 ((𝑋𝑈𝑌𝑉𝑍𝑊) → (2nd ‘⟨𝑋, 𝑌, 𝑍⟩) = 𝑍)
3633, 35sylan9eqr 2676 . . . 4 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (2nd𝐴) = 𝑍)
3729, 32, 363jca 1240 . . 3 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍))
3825, 37jca 554 . 2 (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝐴 = ⟨𝑋, 𝑌, 𝑍⟩) → (𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st𝐴)) = 𝑋 ∧ (2nd ‘(1st𝐴)) = 𝑌 ∧ (2nd𝐴) = 𝑍)))
3915, 38impbida 876 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 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  cop 4174  cotp 4176   × cxp 5102  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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