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

Theorem iunconst 4681
Description: Indexed union of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iunconst (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iunconst
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.9rzv 4209 . . 3 (𝐴 ≠ ∅ → (𝑦𝐵 ↔ ∃𝑥𝐴 𝑦𝐵))
2 eliun 4676 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
31, 2syl6rbbr 279 . 2 (𝐴 ≠ ∅ → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
43eqrdv 2758 1 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wne 2932  wrex 3051  c0 4058   ciun 4672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-nul 4059  df-iun 4674
This theorem is referenced by:  iununi  4762  oe1m  7796  oarec  7813  oelim2  7846  bnj1143  31189  poimirlem32  33772  mblfinlem2  33778  hoicvr  41286  ovnlecvr2  41348  iunhoiioo  41414
  Copyright terms: Public domain W3C validator