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

Theorem eliuni 5001
Description: Membership in an indexed union, one way. (Contributed by JJ, 27-Jul-2021.)
Hypothesis
Ref Expression
eliuni.1 (𝑥 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
eliuni ((𝐴𝐷𝐸𝐶) → 𝐸 𝑥𝐷 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝐸
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem eliuni
StepHypRef Expression
1 eliuni.1 . . . 4 (𝑥 = 𝐴𝐵 = 𝐶)
21eleq2d 2824 . . 3 (𝑥 = 𝐴 → (𝐸𝐵𝐸𝐶))
32rspcev 3621 . 2 ((𝐴𝐷𝐸𝐶) → ∃𝑥𝐷 𝐸𝐵)
4 eliun 4999 . 2 (𝐸 𝑥𝐷 𝐵 ↔ ∃𝑥𝐷 𝐸𝐵)
53, 4sylibr 234 1 ((𝐴𝐷𝐸𝐶) → 𝐸 𝑥𝐷 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wrex 3067   ciun 4995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-v 3479  df-iun 4997
This theorem is referenced by:  oeordi  8623  fseqdom  10063  cfsmolem  10307  axdc3lem2  10488  prmreclem5  16953  efgs1b  19768  lbsextlem2  21178  pmatcoe1fsupp  22722  vitalilem2  25657  weiunse  36450  grpods  42175  oacl2g  43319  omcl2  43322  ofoafg  43343  cnrefiisplem  45784
  Copyright terms: Public domain W3C validator