Theorem relun 4454

Description: The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25. (Contributed by NM, 12-Aug-1994.)

Assertion

Ref	Expression
relun	|- ( Rel ( A u. B ) <-> ( Rel A /\ Rel B ) )

Proof of Theorem relun

Step	Hyp	Ref	Expression
1		unss 3117	. 2 |- ( ( A C_ ( _V X. _V ) /\ B C_ ( _V X. _V ) ) <-> ( A u. B ) C_ ( _V X. _V ) )
2		df-rel 4352	. . 3 |- ( Rel A <-> A C_ ( _V X. _V ) )
3		df-rel 4352	. . 3 |- ( Rel B <-> B C_ ( _V X. _V ) )
4	2, 3	anbi12i 433	. 2 |- ( ( Rel A /\ Rel B ) <-> ( A C_ ( _V X. _V ) /\ B C_ ( _V X. _V ) ) )
5		df-rel 4352	. 2 |- ( Rel ( A u. B ) <-> ( A u. B ) C_ ( _V X. _V ) )
6	1, 4, 5	3bitr4ri 202	1 |- ( Rel ( A u. B ) <-> ( Rel A /\ Rel B ) )

Syntax hints:   /\ wa 97   <-> wb 98   _Vcvv 2557   u. cun 2915   C_ wss 2917   X. cxp 4343   Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-rel 4352

This theorem is referenced by:  funun  4944