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Theorem iunss2 4970
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 4869. (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 4967 . . 3 (∃𝑦𝐵 𝐶𝐷𝐶 𝑦𝐵 𝐷)
21ralimi 3165 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → ∀𝑥𝐴 𝐶 𝑦𝐵 𝐷)
3 iunss 4966 . 2 ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷 ↔ ∀𝑥𝐴 𝐶 𝑦𝐵 𝐷)
42, 3sylibr 235 1 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wral 3143  wrex 3144  wss 3940   ciun 4917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-v 3502  df-in 3947  df-ss 3956  df-iun 4919
This theorem is referenced by:  iunxdif2  4974  oaass  8177  odi  8195  omass  8196  oelim2  8211  cotrclrcl  39952  founiiun  41300  founiiun0  41316  ovnsubaddlem1  42718
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