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 48250
Description: Membership in a union of Cartesian products over its second component, analogous to eliunxp 5848. (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 5703 . . . . . . . 8 Rel (𝐴 × {𝑦})
21rgenw 3065 . . . . . . 7 𝑦𝐵 Rel (𝐴 × {𝑦})
3 reliun 5826 . . . . . . 7 (Rel 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∀𝑦𝐵 Rel (𝐴 × {𝑦}))
42, 3mpbir 231 . . . . . 6 Rel 𝑦𝐵 (𝐴 × {𝑦})
5 elrel 5808 . . . . . 6 ((Rel 𝑦𝐵 (𝐴 × {𝑦}) ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
64, 5mpan 690 . . . . 5 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) → ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
7 excom 2162 . . . . 5 (∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥𝑦 𝐶 = ⟨𝑥, 𝑦⟩)
86, 7sylibr 234 . . . 4 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) → ∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩)
98pm4.71ri 560 . . 3 (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ (∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})))
10 nfiu1 5027 . . . . 5 𝑦 𝑦𝐵 (𝐴 × {𝑦})
1110nfel2 2924 . . . 4 𝑦 𝐶 𝑦𝐵 (𝐴 × {𝑦})
121119.41 2235 . . 3 (∃𝑦(∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ (∃𝑦𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})))
13 19.41v 1949 . . . . 5 (∃𝑥(𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ (∃𝑥 𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})))
14 eleq1 2829 . . . . . . . 8 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦})))
15 opeliun2xp 5753 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑦𝐵𝑥𝐴))
1615biancomi 462 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑥𝐴𝑦𝐵))
1714, 16bitrdi 287 . . . . . . 7 (𝐶 = ⟨𝑥, 𝑦⟩ → (𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑥𝐴𝑦𝐵)))
1817pm5.32i 574 . . . . . 6 ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝐶 𝑦𝐵 (𝐴 × {𝑦})) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
1918exbii 1848 . . . . 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 1848 . . 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 2162 . 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 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  {csn 4626  cop 4632   ciun 4991   × cxp 5683  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-iun 4993  df-opab 5206  df-xp 5691  df-rel 5692
This theorem is referenced by:  mpomptx2  48251
  Copyright terms: Public domain W3C validator