Theorem cnvssb 37892

Description: Subclass theorem for converse. (Contributed by RP, 22-Oct-2020.)

Assertion

Ref	Expression
cnvssb	|- ( Rel A -> ( A C_ B <-> `' A C_ `' B ) )

Proof of Theorem cnvssb

Step	Hyp	Ref	Expression
1		cnvss 5294	. 2 |- ( A C_ B -> `' A C_ `' B )
2		cnvss 5294	. . 3 |- ( `' A C_ `' B -> `' `' A C_ `' `' B )
3		dfrel2 5583	. . . . . . . 8 |- ( Rel A <-> `' `' A = A )
4	3	biimpi 206	. . . . . . 7 |- ( Rel A -> `' `' A = A )
5	4	eqcomd 2628	. . . . . 6 |- ( Rel A -> A = `' `' A )
6	5	adantr 481	. . . . 5 |- ( ( Rel A /\ `' `' A C_ `' `' B ) -> A = `' `' A )
7		id 22	. . . . . . 7 |- ( `' `' A C_ `' `' B -> `' `' A C_ `' `' B )
8		cnvcnvss 5589	. . . . . . 7 |- `' `' B C_ B
9	7, 8	syl6ss 3615	. . . . . 6 |- ( `' `' A C_ `' `' B -> `' `' A C_ B )
10	9	adantl 482	. . . . 5 |- ( ( Rel A /\ `' `' A C_ `' `' B ) -> `' `' A C_ B )
11	6, 10	eqsstrd 3639	. . . 4 |- ( ( Rel A /\ `' `' A C_ `' `' B ) -> A C_ B )
12	11	ex 450	. . 3 |- ( Rel A -> ( `' `' A C_ `' `' B -> A C_ B ) )
13	2, 12	syl5 34	. 2 |- ( Rel A -> ( `' A C_ `' B -> A C_ B ) )
14	1, 13	impbid2 216	1 |- ( Rel A -> ( A C_ B <-> `' A C_ `' B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   C_ wss 3574   `'ccnv 5113   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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  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-rel 5121  df-cnv 5122

This theorem is referenced by: (None)