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Theorem iinuniss 3734
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 2455 . . . 4 ((𝑦𝐴 ∨ ∀𝑥𝐵 𝑦𝑥) → ∀𝑥𝐵 (𝑦𝐴𝑦𝑥))
2 vex 2557 . . . . . 6 𝑦 ∈ V
32elint2 3619 . . . . 5 (𝑦 𝐵 ↔ ∀𝑥𝐵 𝑦𝑥)
43orbi2i 679 . . . 4 ((𝑦𝐴𝑦 𝐵) ↔ (𝑦𝐴 ∨ ∀𝑥𝐵 𝑦𝑥))
5 elun 3081 . . . . 5 (𝑦 ∈ (𝐴𝑥) ↔ (𝑦𝐴𝑦𝑥))
65ralbii 2327 . . . 4 (∀𝑥𝐵 𝑦 ∈ (𝐴𝑥) ↔ ∀𝑥𝐵 (𝑦𝐴𝑦𝑥))
71, 4, 63imtr4i 190 . . 3 ((𝑦𝐴𝑦 𝐵) → ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥))
87ss2abi 3009 . 2 {𝑦 ∣ (𝑦𝐴𝑦 𝐵)} ⊆ {𝑦 ∣ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥)}
9 df-un 2919 . 2 (𝐴 𝐵) = {𝑦 ∣ (𝑦𝐴𝑦 𝐵)}
10 df-iin 3657 . 2 𝑥𝐵 (𝐴𝑥) = {𝑦 ∣ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥)}
118, 9, 103sstr4i 2981 1 (𝐴 𝐵) ⊆ 𝑥𝐵 (𝐴𝑥)
Colors of variables: wff set class
Syntax hints:  wo 629  wcel 1393  {cab 2026  wral 2303  cun 2912  wss 2914   cint 3612   ciin 3655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-int 3613  df-iin 3657
This theorem is referenced by: (None)
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