Theorem iunss2 3853

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 3762. (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 3850	. . 3 ⊢ (∃𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷)
2	1	ralimi 2493	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → ∀𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷)
3		iunss 3849	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷)
4	2, 3	sylibr 133	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4  ∀wral 2414  ∃wrex 2415   ⊆ wss 3066  ∪ ciun 3808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-iun 3810

This theorem is referenced by:  iunxdif2  3856  rdgss  6273