Theorem releccnveq 34022

Description: Equality of converse R-coset and converse S-coset when R and S are relations. (Contributed by Peter Mazsa, 27-Jul-2019.)

Assertion

Ref	Expression
releccnveq	|- ( ( Rel R /\ Rel S ) -> ( [ A ] `' R = [ B ] `' S <-> A. x ( x R A <-> x S B ) ) )

Distinct variable groups:   x, A   x, B   x, R   x, S

Proof of Theorem releccnveq

Step	Hyp	Ref	Expression
1		dfcleq 2616	. 2 |- ( [ A ] `' R = [ B ] `' S <-> A. x ( x e. [ A ] `' R <-> x e. [ B ] `' S ) )
2		releleccnv 34021	. . . 4 |- ( Rel R -> ( x e. [ A ] `' R <-> x R A ) )
3		releleccnv 34021	. . . 4 |- ( Rel S -> ( x e. [ B ] `' S <-> x S B ) )
4	2, 3	bi2bian9 919	. . 3 |- ( ( Rel R /\ Rel S ) -> ( ( x e. [ A ] `' R <-> x e. [ B ] `' S ) <-> ( x R A <-> x S B ) ) )
5	4	albidv 1849	. 2 |- ( ( Rel R /\ Rel S ) -> ( A. x ( x e. [ A ] `' R <-> x e. [ B ] `' S ) <-> A. x ( x R A <-> x S B ) ) )
6	1, 5	syl5bb 272	1 |- ( ( Rel R /\ Rel S ) -> ( [ A ] `' R = [ B ] `' S <-> A. x ( x R A <-> x S B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   class class class wbr 4653   `'ccnv 5113   Rel wrel 5119   [cec 7740

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744

This theorem is referenced by: (None)