Theorem heeq12 38070

Description: Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)

Assertion

Ref	Expression
heeq12	|- ( ( R = S /\ A = B ) -> ( R hereditary A <-> S hereditary B ) )

Proof of Theorem heeq12

Step	Hyp	Ref	Expression
1		simpl 473	. . . 4 |- ( ( R = S /\ A = B ) -> R = S )
2		simpr 477	. . . 4 |- ( ( R = S /\ A = B ) -> A = B )
3	1, 2	imaeq12d 5467	. . 3 |- ( ( R = S /\ A = B ) -> ( R " A ) = ( S " B ) )
4	3, 2	sseq12d 3634	. 2 |- ( ( R = S /\ A = B ) -> ( ( R " A ) C_ A <-> ( S " B ) C_ B ) )
5		df-he 38067	. 2 |- ( R hereditary A <-> ( R " A ) C_ A )
6		df-he 38067	. 2 |- ( S hereditary B <-> ( S " B ) C_ B )
7	4, 5, 6	3bitr4g 303	1 |- ( ( R = S /\ A = B ) -> ( R hereditary A <-> S hereditary B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   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-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-he 38067

This theorem is referenced by:  heeq1  38071  heeq2  38072  frege77  38234