Theorem trclub 13739

Description: The Cartesian product of the domain and range of a relation is an upper bound for its transitive closure. (Contributed by RP, 17-May-2020.)

Assertion

Ref	Expression
trclub	|- ( ( R e. V /\ Rel R ) -> |^| { r | ( R C_ r /\ ( r o. r ) C_ r ) } C_ ( dom R X. ran R ) )

Distinct variable group:   R, r

Allowed substitution hint:   V( r)

Proof of Theorem trclub

Step	Hyp	Ref	Expression
1		relssdmrn 5656	. . . 4 |- ( Rel R -> R C_ ( dom R X. ran R ) )
2		ssequn1 3783	. . . 4 |- ( R C_ ( dom R X. ran R ) <-> ( R u. ( dom R X. ran R ) ) = ( dom R X. ran R ) )
3	1, 2	sylib 208	. . 3 |- ( Rel R -> ( R u. ( dom R X. ran R ) ) = ( dom R X. ran R ) )
4		trclublem 13734	. . 3 |- ( R e. V -> ( R u. ( dom R X. ran R ) ) e. { r | ( R C_ r /\ ( r o. r ) C_ r ) } )
5		eleq1 2689	. . . 4 |- ( ( R u. ( dom R X. ran R ) ) = ( dom R X. ran R ) -> ( ( R u. ( dom R X. ran R ) ) e. { r | ( R C_ r /\ ( r o. r ) C_ r ) } <-> ( dom R X. ran R ) e. { r | ( R C_ r /\ ( r o. r ) C_ r ) } ) )
6	5	biimpa 501	. . 3 |- ( ( ( R u. ( dom R X. ran R ) ) = ( dom R X. ran R ) /\ ( R u. ( dom R X. ran R ) ) e. { r | ( R C_ r /\ ( r o. r ) C_ r ) } ) -> ( dom R X. ran R ) e. { r | ( R C_ r /\ ( r o. r ) C_ r ) } )
7	3, 4, 6	syl2anr 495	. 2 |- ( ( R e. V /\ Rel R ) -> ( dom R X. ran R ) e. { r | ( R C_ r /\ ( r o. r ) C_ r ) } )
8		intss1 4492	. 2 |- ( ( dom R X. ran R ) e. { r | ( R C_ r /\ ( r o. r ) C_ r ) } -> |^| { r | ( R C_ r /\ ( r o. r ) C_ r ) } C_ ( dom R X. ran R ) )
9	7, 8	syl 17	1 |- ( ( R e. V /\ Rel R ) -> |^| { r | ( R C_ r /\ ( r o. r ) C_ r ) } C_ ( dom R X. ran R ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   u. cun 3572   C_ wss 3574   |^|cint 4475   X. cxp 5112   dom cdm 5114   ran crn 5115   o. ccom 5118   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-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-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-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by: (None)