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Theorem iunss2 4721
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 4628. (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 4718 . . 3 (∃𝑦𝐵 𝐶𝐷𝐶 𝑦𝐵 𝐷)
21ralimi 3099 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → ∀𝑥𝐴 𝐶 𝑦𝐵 𝐷)
3 iunss 4717 . 2 ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷 ↔ ∀𝑥𝐴 𝐶 𝑦𝐵 𝐷)
42, 3sylibr 225 1 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wral 3055  wrex 3056  wss 3732   ciun 4676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-v 3352  df-in 3739  df-ss 3746  df-iun 4678
This theorem is referenced by:  iunxdif2  4724  oaass  7846  odi  7864  omass  7865  oelim2  7880  cotrclrcl  38641  founiiun  39939  founiiun0  39956  ovnsubaddlem1  41356
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