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Theorem iinuniss 3902
 Description: A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33 but with equality changed to subset. (Contributed by Jim Kingdon, 19-Aug-2018.)
Assertion
Ref Expression
iinuniss (𝐴 𝐵) ⊆ 𝑥𝐵 (𝐴𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iinuniss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.32vr 2582 . . . 4 ((𝑦𝐴 ∨ ∀𝑥𝐵 𝑦𝑥) → ∀𝑥𝐵 (𝑦𝐴𝑦𝑥))
2 vex 2692 . . . . . 6 𝑦 ∈ V
32elint2 3785 . . . . 5 (𝑦 𝐵 ↔ ∀𝑥𝐵 𝑦𝑥)
43orbi2i 752 . . . 4 ((𝑦𝐴𝑦 𝐵) ↔ (𝑦𝐴 ∨ ∀𝑥𝐵 𝑦𝑥))
5 elun 3221 . . . . 5 (𝑦 ∈ (𝐴𝑥) ↔ (𝑦𝐴𝑦𝑥))
65ralbii 2444 . . . 4 (∀𝑥𝐵 𝑦 ∈ (𝐴𝑥) ↔ ∀𝑥𝐵 (𝑦𝐴𝑦𝑥))
71, 4, 63imtr4i 200 . . 3 ((𝑦𝐴𝑦 𝐵) → ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥))
87ss2abi 3173 . 2 {𝑦 ∣ (𝑦𝐴𝑦 𝐵)} ⊆ {𝑦 ∣ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥)}
9 df-un 3079 . 2 (𝐴 𝐵) = {𝑦 ∣ (𝑦𝐴𝑦 𝐵)}
10 df-iin 3823 . 2 𝑥𝐵 (𝐴𝑥) = {𝑦 ∣ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥)}
118, 9, 103sstr4i 3142 1 (𝐴 𝐵) ⊆ 𝑥𝐵 (𝐴𝑥)
 Colors of variables: wff set class Syntax hints:   ∨ wo 698   ∈ wcel 1481  {cab 2126  ∀wral 2417   ∪ cun 3073   ⊆ wss 3075  ∩ cint 3778  ∩ ciin 3821 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-int 3779  df-iin 3823 This theorem is referenced by: (None)
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