Theorem cotrtrclfv 13753

Description: The transitive closure of a transitive relation. (Contributed by RP, 28-Apr-2020.)

Assertion

Ref	Expression
cotrtrclfv	|- ( ( R e. V /\ ( R o. R ) C_ R ) -> ( t+ ` R ) = R )

Proof of Theorem cotrtrclfv

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		trclfv 13741	. . . 4 |- ( R e. V -> ( t+ ` R ) = |^| { r | ( R C_ r /\ ( r o. r ) C_ r ) } )
2	1	adantr 481	. . 3 |- ( ( R e. V /\ ( R o. R ) C_ R ) -> ( t+ ` R ) = |^| { r | ( R C_ r /\ ( r o. r ) C_ r ) } )
3		simpr 477	. . . . . 6 |- ( ( R e. V /\ ( R o. R ) C_ R ) -> ( R o. R ) C_ R )
4		ssid 3624	. . . . . 6 |- R C_ R
5	3, 4	jctil 560	. . . . 5 |- ( ( R e. V /\ ( R o. R ) C_ R ) -> ( R C_ R /\ ( R o. R ) C_ R ) )
6		trcleq2lem 13730	. . . . . . 7 |- ( r = R -> ( ( R C_ r /\ ( r o. r ) C_ r ) <-> ( R C_ R /\ ( R o. R ) C_ R ) ) )
7	6	elabg 3351	. . . . . 6 |- ( R e. V -> ( R e. { r | ( R C_ r /\ ( r o. r ) C_ r ) } <-> ( R C_ R /\ ( R o. R ) C_ R ) ) )
8	7	adantr 481	. . . . 5 |- ( ( R e. V /\ ( R o. R ) C_ R ) -> ( R e. { r | ( R C_ r /\ ( r o. r ) C_ r ) } <-> ( R C_ R /\ ( R o. R ) C_ R ) ) )
9	5, 8	mpbird 247	. . . 4 |- ( ( R e. V /\ ( R o. R ) C_ R ) -> R e. { r | ( R C_ r /\ ( r o. r ) C_ r ) } )
10		intss1 4492	. . . 4 |- ( R e. { r | ( R C_ r /\ ( r o. r ) C_ r ) } -> |^| { r | ( R C_ r /\ ( r o. r ) C_ r ) } C_ R )
11	9, 10	syl 17	. . 3 |- ( ( R e. V /\ ( R o. R ) C_ R ) -> |^| { r | ( R C_ r /\ ( r o. r ) C_ r ) } C_ R )
12	2, 11	eqsstrd 3639	. 2 |- ( ( R e. V /\ ( R o. R ) C_ R ) -> ( t+ ` R ) C_ R )
13		trclfvlb 13749	. . 3 |- ( R e. V -> R C_ ( t+ ` R ) )
14	13	adantr 481	. 2 |- ( ( R e. V /\ ( R o. R ) C_ R ) -> R C_ ( t+ ` R ) )
15	12, 14	eqssd 3620	1 |- ( ( R e. V /\ ( R o. R ) C_ R ) -> ( t+ ` R ) = R )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   C_ wss 3574   |^|cint 4475   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:  trclidm  13754