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Theorem iunss2 4490
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 4395. (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 4487 . . 3 (∃𝑦𝐵 𝐶𝐷𝐶 𝑦𝐵 𝐷)
21ralimi 2930 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → ∀𝑥𝐴 𝐶 𝑦𝐵 𝐷)
3 iunss 4486 . 2 ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷 ↔ ∀𝑥𝐴 𝐶 𝑦𝐵 𝐷)
42, 3sylibr 222 1 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wral 2890  wrex 2891  wss 3534   ciun 4444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ral 2895  df-rex 2896  df-v 3169  df-in 3541  df-ss 3548  df-iun 4446
This theorem is referenced by:  iunxdif2  4493  oaass  7500  odi  7518  omass  7519  oelim2  7534  cotrclrcl  36851  founiiun  38153  founiiun0  38170  ovnsubaddlem1  39259
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