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

Theorem iinvdif 5080
Description: The indexed intersection of a complement. (Contributed by Gérard Lang, 5-Aug-2018.)
Assertion
Ref Expression
iinvdif 𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iinvdif
StepHypRef Expression
1 dif0 4378 . . . 4 (V ∖ ∅) = V
2 0iun 5063 . . . . 5 𝑥 ∈ ∅ 𝐵 = ∅
32difeq2i 4123 . . . 4 (V ∖ 𝑥 ∈ ∅ 𝐵) = (V ∖ ∅)
4 0iin 5064 . . . 4 𝑥 ∈ ∅ (V ∖ 𝐵) = V
51, 3, 43eqtr4ri 2776 . . 3 𝑥 ∈ ∅ (V ∖ 𝐵) = (V ∖ 𝑥 ∈ ∅ 𝐵)
6 iineq1 5009 . . 3 (𝐴 = ∅ → 𝑥𝐴 (V ∖ 𝐵) = 𝑥 ∈ ∅ (V ∖ 𝐵))
7 iuneq1 5008 . . . 4 (𝐴 = ∅ → 𝑥𝐴 𝐵 = 𝑥 ∈ ∅ 𝐵)
87difeq2d 4126 . . 3 (𝐴 = ∅ → (V ∖ 𝑥𝐴 𝐵) = (V ∖ 𝑥 ∈ ∅ 𝐵))
95, 6, 83eqtr4a 2803 . 2 (𝐴 = ∅ → 𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵))
10 iindif2 5077 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵))
119, 10pm2.61ine 3025 1 𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  Vcvv 3480  cdif 3948  c0 4333   ciun 4991   ciin 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-ss 3968  df-nul 4334  df-iun 4993  df-iin 4994
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator