Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdciin GIF version

Theorem bdciin 13004
Description: The indexed intersection of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
Hypothesis
Ref Expression
bdciun.1 BOUNDED 𝐴
Assertion
Ref Expression
bdciin BOUNDED 𝑥𝑦 𝐴
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem bdciin
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bdciun.1 . . . . 5 BOUNDED 𝐴
21bdeli 12971 . . . 4 BOUNDED 𝑧𝐴
32ax-bdal 12943 . . 3 BOUNDED𝑥𝑦 𝑧𝐴
43bdcab 12974 . 2 BOUNDED {𝑧 ∣ ∀𝑥𝑦 𝑧𝐴}
5 df-iin 3786 . 2 𝑥𝑦 𝐴 = {𝑧 ∣ ∀𝑥𝑦 𝑧𝐴}
64, 5bdceqir 12969 1 BOUNDED 𝑥𝑦 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1465  {cab 2103  wral 2393   ciin 3784  BOUNDED wbdc 12965
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-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-ext 2099  ax-bd0 12938  ax-bdal 12943  ax-bdsb 12947
This theorem depends on definitions:  df-bi 116  df-clab 2104  df-cleq 2110  df-clel 2113  df-iin 3786  df-bdc 12966
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator