Theorem reluni 2493

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(s):   x,A

Proof of Theorem reluni

Step	Hyp	Ref	Expression
1		r19.23v 1282	. . . 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 1923	. . . . 5 |- (y e. U.A <-> E.x e. A y e. x)
3	2	imbi1i 161	. . . 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 154	. . 3 |- (A.x e. A (y e. x -> y e. (V X. V)) <-> (y e. U.A -> y e. (V X. V)))
5	4	bial 695	. 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 2425	. . . . 5 |- (Rel x <-> x (_ (V X. V))
7		dfss2 1497	. . . . 5 |- (x (_ (V X. V) <-> A.y(y e. x -> y e. (V X. V)))
8	6, 7	bitr 151	. . . 4 |- (Rel x <-> A.y(y e. x -> y e. (V X. V)))
9	8	biral 1223	. . 3 |- (A.x e. A Rel x <-> A.x e. A A.y(y e. x -> y e. (V X. V)))
10		ralcom4 1360	. . 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 151	. 2 |- (A.x e. A Rel x <-> A.yA.x e. A (y e. x -> y e. (V X. V)))
12		df-rel 2425	. . 3 |- (Rel U.A <-> U.A (_ (V X. V))
13		dfss2 1497	. . 3 |- (U.A (_ (V X. V) <-> A.y(y e. U.A -> y e. (V X. V)))
14	12, 13	bitr 151	. 2 |- (Rel U.A <-> A.y(y e. U.A -> y e. (V X. V)))
15	5, 11, 14	3bitr4r 159	1 |- (Rel U.A <-> A.x e. A Rel x)

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672   e. wel 803   e. wcel 1092  A.wral 1201  E.wrex 1202  Vcvv 1348   (_ wss 1487  U.cuni 1919   X. cxp 2408  Rel wrel 2415

This theorem is referenced by:  fununi 2705  tfrlem6 2954

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-uni 1920  df-rel 2425

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