MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  el2xptp Structured version   Visualization version   GIF version

Theorem el2xptp 8043
Description: A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
Assertion
Ref Expression
el2xptp (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐷,𝑦,𝑧

Proof of Theorem el2xptp
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 elxp2 5704 . 2 (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧𝐷 𝐴 = ⟨𝑝, 𝑧⟩)
2 opeq1 4876 . . . . 5 (𝑝 = ⟨𝑥, 𝑦⟩ → ⟨𝑝, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
32eqeq2d 2738 . . . 4 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑝, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
43rexbidv 3174 . . 3 (𝑝 = ⟨𝑥, 𝑦⟩ → (∃𝑧𝐷 𝐴 = ⟨𝑝, 𝑧⟩ ↔ ∃𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
54rexxp 5847 . 2 (∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧𝐷 𝐴 = ⟨𝑝, 𝑧⟩ ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
6 df-ot 4639 . . . . . . 7 𝑥, 𝑦, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧
76eqcomi 2736 . . . . . 6 ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨𝑥, 𝑦, 𝑧
87eqeq2i 2740 . . . . 5 (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
98rexbii 3090 . . . 4 (∃𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
109rexbii 3090 . . 3 (∃𝑦𝐶𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
1110rexbii 3090 . 2 (∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
121, 5, 113bitri 296 1 (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥𝐵𝑦𝐶𝑧𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wcel 2098  wrex 3066  cop 4636  cotp 4638   × cxp 5678
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-ot 4639  df-iun 5000  df-opab 5213  df-xp 5686  df-rel 5687
This theorem is referenced by:  el2xpss  8045  ralxp3f  8146  frpoins3xp3g  8150  poxp3  8159  xpord3pred  8161  sexp3  8162
  Copyright terms: Public domain W3C validator