Theorem iunss2 4012

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 3923. (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 4009	. . 3 ⊢ (∃y ∈ B C ⊆ D → C ⊆ ∪y ∈ B D)
2	1	ralimi 2690	. 2 ⊢ (∀x ∈ A ∃y ∈ B C ⊆ D → ∀x ∈ A C ⊆ ∪y ∈ B D)
3		iunss 4008	. 2 ⊢ (∪x ∈ A C ⊆ ∪y ∈ B D ↔ ∀x ∈ A C ⊆ ∪y ∈ B D)
4	2, 3	sylibr 203	1 ⊢ (∀x ∈ A ∃y ∈ B C ⊆ D → ∪x ∈ A C ⊆ ∪y ∈ B D)

Syntax hints:   → wi 4  ∀wral 2615  ∃wrex 2616   ⊆ wss 3258  ∪ciun 3970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-iun 3972

This theorem is referenced by:  iunxdif2  4015