Theorem iunss2 2599

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 2533.

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 groups:   x,y   x,B   y,C   x,D

Proof of Theorem iunss2

Step	Hyp	Ref	Expression
1		ssiun 2596	. . 3 |- (E.y e. B C (_ D -> C (_ U_y e. B D)
2	1	r19.20si 1709	. 2 |- (A.x e. A E.y e. B C (_ D -> A.x e. A C (_ U_y e. B D)
3		iunss 2595	. 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 200	1 |- (A.x e. A E.y e. B C (_ D -> U_x e. A C (_ U_y e. B D)

Syntax hints:   -> wi 3  A.wral 1648  E.wrex 1649   (_ wss 2050  U_ciun 2570

This theorem is referenced by:  iunxdif2 2602  oaass 4201  odi 4216  omass 4217  oelim2 4228

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815  df-in 2054  df-ss 2056  df-iun 2572

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