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Theorem iunss2 4945
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 4841. (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 4942 . . 3 (∃𝑦𝐵 𝐶𝐷𝐶 𝑦𝐵 𝐷)
21ralimi 3076 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → ∀𝑥𝐴 𝐶 𝑦𝐵 𝐷)
3 iunss 4941 . 2 ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷 ↔ ∀𝑥𝐴 𝐶 𝑦𝐵 𝐷)
42, 3sylibr 237 1 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wral 3054  wrex 3055  wss 3853   ciun 4891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-v 3402  df-in 3860  df-ss 3870  df-iun 4893
This theorem is referenced by:  iunxdif2  4949  oaass  8231  odi  8249  omass  8250  oelim2  8265  cotrclrcl  40937  founiiun  42294  founiiun0  42307  ovnsubaddlem1  43691
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