Theorem reluni 3255

Description: Union law for relations. Exercise 6 of [TakeutiZaring] p. 25 and its converse.

Assertion

Ref	Expression
reluni	|- (Rel U.A <-> A.x e. A Rel x)

Distinct variable group:   x,A

Proof of Theorem reluni

Step	Hyp	Ref	Expression
1		r19.23v 1733	. . . 4 |- (A.x e. A (y e. x -> y e. (V X. V)) <-> (E.x e. A y e. x -> y e. (V X. V)))
2		eluni2 2497	. . . . 5 |- (y e. U.A <-> E.x e. A y e. x)
3	2	imbi1i 186	. . . 4 |- ((y e. U.A -> y e. (V X. V)) <-> (E.x e. A y e. x -> y e. (V X. V)))
4	1, 3	bitr4 176	. . 3 |- (A.x e. A (y e. x -> y e. (V X. V)) <-> (y e. U.A -> y e. (V X. V)))
5	4	albii 996	. 2 |- (A.yA.x e. A (y e. x -> y e. (V X. V)) <-> A.y(y e. U.A -> y e. (V X. V)))
6		df-rel 3175	. . . . 5 |- (Rel x <-> x (_ (V X. V))
7		dfss2 2048	. . . . 5 |- (x (_ (V X. V) <-> A.y(y e. x -> y e. (V X. V)))
8	6, 7	bitr 173	. . . 4 |- (Rel x <-> A.y(y e. x -> y e. (V X. V)))
9	8	ralbii 1659	. . 3 |- (A.x e. A Rel x <-> A.x e. A A.y(y e. x -> y e. (V X. V)))
10		ralcom4 1814	. . 3 |- (A.x e. A A.y(y e. x -> y e. (V X. V)) <-> A.yA.x e. A (y e. x -> y e. (V X. V)))
11	9, 10	bitr 173	. 2 |- (A.x e. A Rel x <-> A.yA.x e. A (y e. x -> y e. (V X. V)))
12		df-rel 3175	. . 3 |- (Rel U.A <-> U.A (_ (V X. V))
13		dfss2 2048	. . 3 |- (U.A (_ (V X. V) <-> A.y(y e. U.A -> y e. (V X. V)))
14	12, 13	bitr 173	. 2 |- (Rel U.A <-> A.y(y e. U.A -> y e. (V X. V)))
15	5, 11, 14	3bitr4r 184	1 |- (Rel U.A <-> A.x e. A Rel x)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 951   e. wcel 955  A.wral 1637  E.wrex 1638  Vcvv 1802   (_ wss 2037  U.cuni 2493   X. cxp 3158  Rel wrel 3165

This theorem is referenced by:  fununi 3549  tfrlem6 3901

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-rex 1642  df-v 1803  df-in 2041  df-ss 2043  df-uni 2494  df-rel 3175

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