Theorem iunss2 2021

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 1942.

Assertion

Ref	Expression
iunss2	|- (A.x e. A E.y e. B C (_ D -> U.x e. A C (_ U.y e. B D)

Distinct variable group(s):   x,y   x,B   y,C   x,D

Proof of Theorem iunss2

Step	Hyp	Ref	Expression
1		ssiun 2018	. . 3 |- (E.y e. B C (_ D -> C (_ U.y e. B D)
2	1	r19.20si 1254	. 2 |- (A.x e. A E.y e. B C (_ D -> A.x e. A C (_ U.y e. B D)
3		iunss 2017	. 2 |- (U.x e. A C (_ U.y e. B D <-> A.x e. A C (_ U.y e. B D)
4	2, 3	sylibr 175	1 |- (A.x e. A E.y e. B C (_ D -> U.x e. A C (_ U.y e. B D)

Syntax hints:   -> wi 2  A.wral 1201  E.wrex 1202   (_ wss 1487  U.ciun 1994

This theorem is referenced by:  oaass 3163

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-in 1491  df-ss 1492  df-iun 1996

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