Theorem trclfv 13741

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

Assertion

Ref	Expression
trclfv	|- ( R e. V -> ( t+ ` R ) = |^| { x | ( R C_ x /\ ( x o. x ) C_ x ) } )

Distinct variable group:   x, R

Allowed substitution hint:   V( x)

Proof of Theorem trclfv

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. . 3 |- ( R e. V -> R e. _V )
2		trclexlem 13733	. . . 4 |- ( R e. _V -> ( R u. ( dom R X. ran R ) ) e. _V )
3		trclubg 13740	. . . 4 |- ( R e. _V -> |^| { x | ( R C_ x /\ ( x o. x ) C_ x ) } C_ ( R u. ( dom R X. ran R ) ) )
4	2, 3	ssexd 4805	. . 3 |- ( R e. _V -> |^| { x | ( R C_ x /\ ( x o. x ) C_ x ) } e. _V )
5	1, 4	jccir 562	. 2 |- ( R e. V -> ( R e. _V /\ |^| { x | ( R C_ x /\ ( x o. x ) C_ x ) } e. _V ) )
6		trcleq1 13728	. . 3 |- ( r = R -> |^| { x | ( r C_ x /\ ( x o. x ) C_ x ) } = |^| { x | ( R C_ x /\ ( x o. x ) C_ x ) } )
7		df-trcl 13726	. . 3 |- t+ = ( r e. _V |-> |^| { x | ( r C_ x /\ ( x o. x ) C_ x ) } )
8	6, 7	fvmptg 6280	. 2 |- ( ( R e. _V /\ |^| { x | ( R C_ x /\ ( x o. x ) C_ x ) } e. _V ) -> ( t+ ` R ) = |^| { x | ( R C_ x /\ ( x o. x ) C_ x ) } )
9	5, 8	syl 17	1 |- ( R e. V -> ( t+ ` R ) = |^| { x | ( R C_ x /\ ( x o. x ) C_ x ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   u. cun 3572   C_ wss 3574   |^|cint 4475   X. cxp 5112   dom cdm 5114   ran crn 5115   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:  brtrclfv  13743  trclfvss  13747  trclfvub  13748  trclfvlb  13749  cotrtrclfv  13753  trclun  13755  brtrclfv2  38019