Theorem iunss2 5016

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 4908. (Contributed by NM, 9-Dec-2004.)

Assertion

Ref	Expression
iunss2	⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 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

Step	Hyp	Ref	Expression
1		ssiun 5013	. . 3 ⊢ (∃𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷)
2	1	ralimi 3067	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → ∀𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷)
3		iunss 5012	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷)
4	2, 3	sylibr 234	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4  ∀wral 3045  ∃wrex 3054   ⊆ wss 3917  ∪ ciun 4958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-v 3452  df-ss 3934  df-iun 4960

This theorem is referenced by:  iunxdif2  5020  oaass  8528  odi  8546  omass  8547  oelim2  8562  cotrclrcl  43738  founiiun  45180  founiiun0  45191  ovnsubaddlem1  46575