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

Theorem iinconst 4920
 Description: Indexed intersection of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by Mario Carneiro, 6-Feb-2015.)
Assertion
Ref Expression
iinconst (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iinconst
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.3rzv 4442 . . 3 (𝐴 ≠ ∅ → (𝑦𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
2 eliin 4915 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
32elv 3498 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
41, 3syl6rbbr 292 . 2 (𝐴 ≠ ∅ → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
54eqrdv 2817 1 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   = wceq 1530   ∈ wcel 2107   ≠ wne 3014  ∀wral 3136  Vcvv 3493  ∅c0 4289  ∩ ciin 4911 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-v 3495  df-dif 3937  df-nul 4290  df-iin 4913 This theorem is referenced by:  iin0  5252  ptbasfi  22181
 Copyright terms: Public domain W3C validator