Theorem unielrel 5660

Description: The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)

Assertion

Ref	Expression
unielrel	|- ( ( Rel R /\ A e. R ) -> U. A e. U. R )

Proof of Theorem unielrel

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elrel 5222	. 2 |- ( ( Rel R /\ A e. R ) -> E. x E. y A = <. x , y >. )
2		simpr 477	. 2 |- ( ( Rel R /\ A e. R ) -> A e. R )
3		vex 3203	. . . . . 6 |- x e. _V
4		vex 3203	. . . . . 6 |- y e. _V
5	3, 4	uniopel 4978	. . . . 5 |- ( <. x , y >. e. R -> U. <. x , y >. e. U. R )
6	5	a1i 11	. . . 4 |- ( A = <. x , y >. -> ( <. x , y >. e. R -> U. <. x , y >. e. U. R ) )
7		eleq1 2689	. . . 4 |- ( A = <. x , y >. -> ( A e. R <-> <. x , y >. e. R ) )
8		unieq 4444	. . . . 5 |- ( A = <. x , y >. -> U. A = U. <. x , y >. )
9	8	eleq1d 2686	. . . 4 |- ( A = <. x , y >. -> ( U. A e. U. R <-> U. <. x , y >. e. U. R ) )
10	6, 7, 9	3imtr4d 283	. . 3 |- ( A = <. x , y >. -> ( A e. R -> U. A e. U. R ) )
11	10	exlimivv 1860	. 2 |- ( E. x E. y A = <. x , y >. -> ( A e. R -> U. A e. U. R ) )
12	1, 2, 11	sylc 65	1 |- ( ( Rel R /\ A e. R ) -> U. A e. U. R )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   <.cop 4183   U.cuni 4436   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by: (None)