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

Theorem iunconst 4920
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 4444 . . 3 (𝐴 ≠ ∅ → (𝑦𝐵 ↔ ∃𝑥𝐴 𝑦𝐵))
2 eliun 4915 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
31, 2syl6rbbr 292 . 2 (𝐴 ≠ ∅ → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
43eqrdv 2819 1 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2110  wne 3016  wrex 3139  c0 4290   ciun 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-v 3496  df-dif 3938  df-nul 4291  df-iun 4913
This theorem is referenced by:  iununi  5013  oe1m  8165  oarec  8182  oelim2  8215  bnj1143  32057  poimirlem32  34918  mblfinlem2  34924  scottrankd  40577  hoicvr  42824  ovnlecvr2  42886  iunhoiioo  42952
  Copyright terms: Public domain W3C validator