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Theorem iinvdif 4528
 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 3904 . . . 4 (V ∖ ∅) = V
2 0iun 4513 . . . . 5 𝑥 ∈ ∅ 𝐵 = ∅
32difeq2i 3687 . . . 4 (V ∖ 𝑥 ∈ ∅ 𝐵) = (V ∖ ∅)
4 0iin 4514 . . . 4 𝑥 ∈ ∅ (V ∖ 𝐵) = V
51, 3, 43eqtr4ri 2643 . . 3 𝑥 ∈ ∅ (V ∖ 𝐵) = (V ∖ 𝑥 ∈ ∅ 𝐵)
6 iineq1 4471 . . 3 (𝐴 = ∅ → 𝑥𝐴 (V ∖ 𝐵) = 𝑥 ∈ ∅ (V ∖ 𝐵))
7 iuneq1 4470 . . . 4 (𝐴 = ∅ → 𝑥𝐴 𝐵 = 𝑥 ∈ ∅ 𝐵)
87difeq2d 3690 . . 3 (𝐴 = ∅ → (V ∖ 𝑥𝐴 𝐵) = (V ∖ 𝑥 ∈ ∅ 𝐵))
95, 6, 83eqtr4a 2670 . 2 (𝐴 = ∅ → 𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵))
10 iindif2 4525 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵))
119, 10pm2.61ine 2865 1 𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  ∪ ciun 4455  ∩ ciin 4456 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-iun 4457  df-iin 4458 This theorem is referenced by: (None)
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