Theorem iunxdif2 2602

Description: Indexed union with a class difference as its index.

Hypothesis

Ref	Expression
iunxdif2.1	|- (x = y -> C = D)

Assertion

Ref	Expression
iunxdif2	|- (A.x e. A E.y e. (A \ B)C (_ D -> U_y e. (A \ B)D = U_x e. A C)

Distinct variable groups:   x,y,A   x,B,y   y,C   x,D

Proof of Theorem iunxdif2

Step	Hyp	Ref	Expression
1		iunss2 2599	. . 3 |- (A.x e. A E.y e. (A \ B)C (_ D -> U_x e. A C (_ U_y e. (A \ B)D)
2		difss 2170	. . . . 5 |- (A \ B) (_ A
3		iunss1 2578	. . . . 5 |- ((A \ B) (_ A -> U_y e. (A \ B)D (_ U_y e. A D)
4	2, 3	ax-mp 7	. . . 4 |- U_y e. (A \ B)D (_ U_y e. A D
5		iunxdif2.1	. . . . 5 |- (x = y -> C = D)
6	5	cbviunv 2594	. . . 4 |- U_x e. A C = U_y e. A D
7	4, 6	sseqtr4 2097	. . 3 |- U_y e. (A \ B)D (_ U_x e. A C
8	1, 7	jctil 292	. 2 |- (A.x e. A E.y e. (A \ B)C (_ D -> (U_y e. (A \ B)D (_ U_x e. A C /\ U_x e. A C (_ U_y e. (A \ B)D))
9		eqss 2080	. 2 |- (U_y e. (A \ B)D = U_x e. A C <-> (U_y e. (A \ B)D (_ U_x e. A C /\ U_x e. A C (_ U_y e. (A \ B)D))
10	8, 9	sylibr 200	1 |- (A.x e. A E.y e. (A \ B)C (_ D -> U_y e. (A \ B)D = U_x e. A C)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958  A.wral 1648  E.wrex 1649   \ cdif 2047   (_ wss 2050  U_ciun 2570

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-dif 2052  df-in 2054  df-ss 2056  df-iun 2572

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