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Theorem iinuniss 3987
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 2638 . . . 4 ((𝑦𝐴 ∨ ∀𝑥𝐵 𝑦𝑥) → ∀𝑥𝐵 (𝑦𝐴𝑦𝑥))
2 vex 2755 . . . . . 6 𝑦 ∈ V
32elint2 3869 . . . . 5 (𝑦 𝐵 ↔ ∀𝑥𝐵 𝑦𝑥)
43orbi2i 763 . . . 4 ((𝑦𝐴𝑦 𝐵) ↔ (𝑦𝐴 ∨ ∀𝑥𝐵 𝑦𝑥))
5 elun 3291 . . . . 5 (𝑦 ∈ (𝐴𝑥) ↔ (𝑦𝐴𝑦𝑥))
65ralbii 2496 . . . 4 (∀𝑥𝐵 𝑦 ∈ (𝐴𝑥) ↔ ∀𝑥𝐵 (𝑦𝐴𝑦𝑥))
71, 4, 63imtr4i 201 . . 3 ((𝑦𝐴𝑦 𝐵) → ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥))
87ss2abi 3242 . 2 {𝑦 ∣ (𝑦𝐴𝑦 𝐵)} ⊆ {𝑦 ∣ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥)}
9 df-un 3148 . 2 (𝐴 𝐵) = {𝑦 ∣ (𝑦𝐴𝑦 𝐵)}
10 df-iin 3907 . 2 𝑥𝐵 (𝐴𝑥) = {𝑦 ∣ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥)}
118, 9, 103sstr4i 3211 1 (𝐴 𝐵) ⊆ 𝑥𝐵 (𝐴𝑥)
Colors of variables: wff set class
Syntax hints:  wo 709  wcel 2160  {cab 2175  wral 2468  cun 3142  wss 3144   cint 3862   ciin 3905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-int 3863  df-iin 3907
This theorem is referenced by: (None)
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