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Theorem iinvdif 4966
 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 4286 . . . 4 (V ∖ ∅) = V
2 0iun 4950 . . . . 5 𝑥 ∈ ∅ 𝐵 = ∅
32difeq2i 4047 . . . 4 (V ∖ 𝑥 ∈ ∅ 𝐵) = (V ∖ ∅)
4 0iin 4951 . . . 4 𝑥 ∈ ∅ (V ∖ 𝐵) = V
51, 3, 43eqtr4ri 2832 . . 3 𝑥 ∈ ∅ (V ∖ 𝐵) = (V ∖ 𝑥 ∈ ∅ 𝐵)
6 iineq1 4899 . . 3 (𝐴 = ∅ → 𝑥𝐴 (V ∖ 𝐵) = 𝑥 ∈ ∅ (V ∖ 𝐵))
7 iuneq1 4898 . . . 4 (𝐴 = ∅ → 𝑥𝐴 𝐵 = 𝑥 ∈ ∅ 𝐵)
87difeq2d 4050 . . 3 (𝐴 = ∅ → (V ∖ 𝑥𝐴 𝐵) = (V ∖ 𝑥 ∈ ∅ 𝐵))
95, 6, 83eqtr4a 2859 . 2 (𝐴 = ∅ → 𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵))
10 iindif2 4963 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵))
119, 10pm2.61ine 3070 1 𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  Vcvv 3441   ∖ cdif 3878  ∅c0 4243  ∪ ciun 4882  ∩ ciin 4883 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-in 3888  df-ss 3898  df-nul 4244  df-iun 4884  df-iin 4885 This theorem is referenced by: (None)
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