MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunss2 Structured version   Visualization version   GIF version

Theorem iunss2 4699
Description: A subclass condition on the members of two indexed classes 𝐶(𝑥) and 𝐷(𝑦) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 4606. (Contributed by NM, 9-Dec-2004.)
Assertion
Ref Expression
iunss2 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐵   𝑦,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦)   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem iunss2
StepHypRef Expression
1 ssiun 4696 . . 3 (∃𝑦𝐵 𝐶𝐷𝐶 𝑦𝐵 𝐷)
21ralimi 3101 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → ∀𝑥𝐴 𝐶 𝑦𝐵 𝐷)
3 iunss 4695 . 2 ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷 ↔ ∀𝑥𝐴 𝐶 𝑦𝐵 𝐷)
42, 3sylibr 224 1 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wral 3061  wrex 3062  wss 3723   ciun 4654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-in 3730  df-ss 3737  df-iun 4656
This theorem is referenced by:  iunxdif2  4702  oaass  7795  odi  7813  omass  7814  oelim2  7829  cotrclrcl  38560  founiiun  39880  founiiun0  39897  ovnsubaddlem1  41304
  Copyright terms: Public domain W3C validator