Theorem brtrclfv 13743

Description: Two ways of expressing the transitive closure of a binary relation. (Contributed by RP, 9-May-2020.)

Assertion

Ref	Expression
brtrclfv	|- ( R e. V -> ( A ( t+ ` R ) B <-> A. r ( ( R C_ r /\ ( r o. r ) C_ r ) -> A r B ) ) )

Distinct variable groups:   A, r   B, r   R, r

Allowed substitution hint:   V( r)

Proof of Theorem brtrclfv

Step	Hyp	Ref	Expression
1		trclfv 13741	. . 3 |- ( R e. V -> ( t+ ` R ) = |^| { r | ( R C_ r /\ ( r o. r ) C_ r ) } )
2	1	breqd 4664	. 2 |- ( R e. V -> ( A ( t+ ` R ) B <-> A |^| { r | ( R C_ r /\ ( r o. r ) C_ r ) } B ) )
3		brintclab 13742	. . 3 |- ( A |^| { r | ( R C_ r /\ ( r o. r ) C_ r ) } B <-> A. r ( ( R C_ r /\ ( r o. r ) C_ r ) -> <. A , B >. e. r ) )
4		df-br 4654	. . . . 5 |- ( A r B <-> <. A , B >. e. r )
5	4	imbi2i 326	. . . 4 |- ( ( ( R C_ r /\ ( r o. r ) C_ r ) -> A r B ) <-> ( ( R C_ r /\ ( r o. r ) C_ r ) -> <. A , B >. e. r ) )
6	5	albii 1747	. . 3 |- ( A. r ( ( R C_ r /\ ( r o. r ) C_ r ) -> A r B ) <-> A. r ( ( R C_ r /\ ( r o. r ) C_ r ) -> <. A , B >. e. r ) )
7	3, 6	bitr4i 267	. 2 |- ( A |^| { r | ( R C_ r /\ ( r o. r ) C_ r ) } B <-> A. r ( ( R C_ r /\ ( r o. r ) C_ r ) -> A r B ) )
8	2, 7	syl6bb 276	1 |- ( R e. V -> ( A ( t+ ` R ) B <-> A. r ( ( R C_ r /\ ( r o. r ) C_ r ) -> A r B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   {cab 2608   C_ wss 3574   <.cop 4183   |^|cint 4475   class class class wbr 4653   o. ccom 5118   ` cfv 5888   t+ctcl 13724

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-ne 2795  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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-trcl 13726

This theorem is referenced by:  brcnvtrclfv  13744  brtrclfvcnv  13745  trclfvcotr  13750