Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eliunxp2 Structured version   Visualization version   GIF version

Theorem eliunxp2 46483
Description: Membership in a union of Cartesian products over its second component, analogous to eliunxp 5798. (Contributed by AV, 30-Mar-2019.)
Assertion
Ref Expression
eliunxp2 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑦,𝐶
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑦)

Proof of Theorem eliunxp2
StepHypRef Expression
1 relxp 5656 . . . . . . . 8 Rel (𝐴 × {𝑦})
21rgenw 3069 . . . . . . 7 𝑦𝐵 Rel (𝐴 × {𝑦})
3 reliun 5777 . . . . . . 7 (Rel 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∀𝑦𝐵 Rel (𝐴 × {𝑦}))
42, 3mpbir 230 . . . . . 6 Rel 𝑦𝐵 (𝐴 × {𝑦})
5 elrel 5759 . . . . . 6 ((Rel 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
64, 5mpan 689 . . . . 5 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
7 excom 2163 . . . . 5 (∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
86, 7sylibr 233 . . . 4 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) → ∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩)
98pm4.71ri 562 . . 3 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ (∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})))
10 nfiu1 4993 . . . . 5 𝑦 𝑦𝐵 (𝐴 × {𝑦})
1110nfel2 2926 . . . 4 𝑦 𝐶 𝑦𝐵 (𝐴 × {𝑦})
121119.41 2229 . . 3 (∃𝑦(∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ (∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})))
13 19.41v 1954 . . . . 5 (∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ (∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})))
14 eleq1 2826 . . . . . . . 8 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦})))
15 opeliun2xp 46482 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑦𝐵𝑥𝐴))
1615biancomi 464 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑥𝐴𝑦𝐵))
1714, 16bitrdi 287 . . . . . . 7 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑥𝐴𝑦𝐵)))
1817pm5.32i 576 . . . . . 6 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
1918exbii 1851 . . . . 5 (∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ ∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
2013, 19bitr3i 277 . . . 4 ((∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ ∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
2120exbii 1851 . . 3 (∃𝑦(∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ ∃𝑦𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
229, 12, 213bitr2i 299 . 2 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∃𝑦𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
23 excom 2163 . 2 (∃𝑦𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
2422, 23bitri 275 1 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wral 3065  {csn 4591  cop 4597   ciun 4959   × cxp 5636  Rel wrel 5643
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-iun 4961  df-opab 5173  df-xp 5644  df-rel 5645
This theorem is referenced by:  mpomptx2  46484
  Copyright terms: Public domain W3C validator