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Theorem iinvdif 4522
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 3903 . . . 4 (V ∖ ∅) = V
2 0iun 4507 . . . . 5 𝑥 ∈ ∅ 𝐵 = ∅
32difeq2i 3686 . . . 4 (V ∖ 𝑥 ∈ ∅ 𝐵) = (V ∖ ∅)
4 0iin 4508 . . . 4 𝑥 ∈ ∅ (V ∖ 𝐵) = V
51, 3, 43eqtr4ri 2642 . . 3 𝑥 ∈ ∅ (V ∖ 𝐵) = (V ∖ 𝑥 ∈ ∅ 𝐵)
6 iineq1 4465 . . 3 (𝐴 = ∅ → 𝑥𝐴 (V ∖ 𝐵) = 𝑥 ∈ ∅ (V ∖ 𝐵))
7 iuneq1 4464 . . . 4 (𝐴 = ∅ → 𝑥𝐴 𝐵 = 𝑥 ∈ ∅ 𝐵)
87difeq2d 3689 . . 3 (𝐴 = ∅ → (V ∖ 𝑥𝐴 𝐵) = (V ∖ 𝑥 ∈ ∅ 𝐵))
95, 6, 83eqtr4a 2669 . 2 (𝐴 = ∅ → 𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵))
10 iindif2 4519 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵))
119, 10pm2.61ine 2864 1 𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  Vcvv 3172  cdif 3536  c0 3873   ciun 4449   ciin 4450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-in 3546  df-ss 3553  df-nul 3874  df-iun 4451  df-iin 4452
This theorem is referenced by: (None)
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