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

Theorem iinconst 4720
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 4257 . . 3 (𝐴 ≠ ∅ → (𝑦𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
2 vex 3388 . . . 4 𝑦 ∈ V
3 eliin 4715 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
42, 3ax-mp 5 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
51, 4syl6rbbr 282 . 2 (𝐴 ≠ ∅ → (𝑦 𝑥𝐴 𝐵𝑦𝐵))
65eqrdv 2797 1 (𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  wcel 2157  wne 2971  wral 3089  Vcvv 3385  c0 4115   ciin 4711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-v 3387  df-dif 3772  df-nul 4116  df-iin 4713
This theorem is referenced by:  iin0  5031  ptbasfi  21713
  Copyright terms: Public domain W3C validator