Theorem unhe1 38079

Description: The union of two relations hereditary in a class is also hereditary in a class. (Contributed by RP, 28-Mar-2020.)

Assertion

Ref	Expression
unhe1	|- ( ( R hereditary A /\ S hereditary A ) -> ( R u. S ) hereditary A )

Proof of Theorem unhe1

Step	Hyp	Ref	Expression
1		df-he 38067	. . 3 |- ( R hereditary A <-> ( R " A ) C_ A )
2		df-he 38067	. . 3 |- ( S hereditary A <-> ( S " A ) C_ A )
3		imaundir 5546	. . . 4 |- ( ( R u. S ) " A ) = ( ( R " A ) u. ( S " A ) )
4		unss 3787	. . . . 5 |- ( ( ( R " A ) C_ A /\ ( S " A ) C_ A ) <-> ( ( R " A ) u. ( S " A ) ) C_ A )
5	4	biimpi 206	. . . 4 |- ( ( ( R " A ) C_ A /\ ( S " A ) C_ A ) -> ( ( R " A ) u. ( S " A ) ) C_ A )
6	3, 5	syl5eqss 3649	. . 3 |- ( ( ( R " A ) C_ A /\ ( S " A ) C_ A ) -> ( ( R u. S ) " A ) C_ A )
7	1, 2, 6	syl2anb 496	. 2 |- ( ( R hereditary A /\ S hereditary A ) -> ( ( R u. S ) " A ) C_ A )
8		df-he 38067	. 2 |- ( ( R u. S ) hereditary A <-> ( ( R u. S ) " A ) C_ A )
9	7, 8	sylibr 224	1 |- ( ( R hereditary A /\ S hereditary A ) -> ( R u. S ) hereditary A )

Syntax hints:   -> wi 4   /\ wa 384   u. cun 3572   C_ wss 3574   "cima 5117   hereditary whe 38066

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

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-rab 2921  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-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-he 38067

This theorem is referenced by:  sshepw  38083