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Theorem trclubgNEW 37925
Description: If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.)
Hypothesis
Ref Expression
trclubgNEW.rex |- ( ph -> R e. _V )
Assertion
Ref Expression
trclubgNEW |- ( ph -> |^| { x | ( R C_ x /\ ( x o. x ) C_ x ) } C_ ( R u. ( dom R X. ran R ) ) )
Distinct variable group:   x, R
Allowed substitution hint:   ph( x)
Proof of Theorem trclubgNEW
StepHypRef Expression
1 trclubgNEW.rex . . 3 |- ( ph -> R e. _V )
2 dmexg 7097 . . . . 5 |- ( R e. _V -> dom R e. _V )
31, 2syl 17 . . . 4 |- ( ph -> dom R e. _V )
4 rnexg 7098 . . . . 5 |- ( R e. _V -> ran R e. _V )
51, 4syl 17 . . . 4 |- ( ph -> ran R e. _V )
6 xpexg 6960 . . . 4 |- ( ( dom R e. _V /\ ran R e. _V ) -> ( dom R X. ran R ) e. _V )
73, 5, 6syl2anc 693 . . 3 |- ( ph -> ( dom R X. ran R ) e. _V )
8 unexg 6959 . . 3 |- ( ( R e. _V /\ ( dom R X. ran R ) e. _V ) -> ( R u. ( dom R X. ran R ) ) e. _V )
91, 7, 8syl2anc 693 . 2 |- ( ph -> ( R u. ( dom R X. ran R ) ) e. _V )
10 id 22 . . . 4 |- ( x = ( R u. ( dom R X. ran R ) ) -> x = ( R u. ( dom R X. ran R ) ) )
1110, 10coeq12d 5286 . . 3 |- ( x = ( R u. ( dom R X. ran R ) ) -> ( x o. x ) = ( ( R u. ( dom R X. ran R ) ) o. ( R u. ( dom R X. ran R ) ) ) )
1211, 10sseq12d 3634 . 2 |- ( x = ( R u. ( dom R X. ran R ) ) -> ( ( x o. x ) C_ x <-> ( ( R u. ( dom R X. ran R ) ) o. ( R u. ( dom R X. ran R ) ) ) C_ ( R u. ( dom R X. ran R ) ) ) )
13 ssun1 3776 . . 3 |- R C_ ( R u. ( dom R X. ran R ) )
1413a1i 11 . 2 |- ( ph -> R C_ ( R u. ( dom R X. ran R ) ) )
15 cnvssrndm 5657 . . 3 |- `' R C_ ( ran R X. dom R )
16 coundi 5636 . . . 4 |- ( ( R u. ( dom R X. ran R ) ) o. ( R u. ( dom R X. ran R ) ) ) = ( ( ( R u. ( dom R X. ran R ) ) o. R ) u. ( ( R u. ( dom R X. ran R ) ) o. ( dom R X. ran R ) ) )
17 cnvss 5294 . . . . . . . 8 |- ( `' R C_ ( ran R X. dom R ) -> `' `' R C_ `' ( ran R X. dom R ) )
18 coss2 5278 . . . . . . . 8 |- ( `' `' R C_ `' ( ran R X. dom R ) -> ( ( R u. ( dom R X. ran R ) ) o. `' `' R ) C_ ( ( R u. ( dom R X. ran R ) ) o. `' ( ran R X. dom R ) ) )
1917, 18syl 17 . . . . . . 7 |- ( `' R C_ ( ran R X. dom R ) -> ( ( R u. ( dom R X. ran R ) ) o. `' `' R ) C_ ( ( R u. ( dom R X. ran R ) ) o. `' ( ran R X. dom R ) ) )
20 cocnvcnv2 5647 . . . . . . 7 |- ( ( R u. ( dom R X. ran R ) ) o. `' `' R ) = ( ( R u. ( dom R X. ran R ) ) o. R )
21 cnvxp 5551 . . . . . . . 8 |- `' ( ran R X. dom R ) = ( dom R X. ran R )
2221coeq2i 5282 . . . . . . 7 |- ( ( R u. ( dom R X. ran R ) ) o. `' ( ran R X. dom R ) ) = ( ( R u. ( dom R X. ran R ) ) o. ( dom R X. ran R ) )
2319, 20, 223sstr3g 3645 . . . . . 6 |- ( `' R C_ ( ran R X. dom R ) -> ( ( R u. ( dom R X. ran R ) ) o. R ) C_ ( ( R u. ( dom R X. ran R ) ) o. ( dom R X. ran R ) ) )
24 ssequn1 3783 . . . . . 6 |- ( ( ( R u. ( dom R X. ran R ) ) o. R ) C_ ( ( R u. ( dom R X. ran R ) ) o. ( dom R X. ran R ) ) <-> ( ( ( R u. ( dom R X. ran R ) ) o. R ) u. ( ( R u. ( dom R X. ran R ) ) o. ( dom R X. ran R ) ) ) = ( ( R u. ( dom R X. ran R ) ) o. ( dom R X. ran R ) ) )
2523, 24sylib 208 . . . . 5 |- ( `' R C_ ( ran R X. dom R ) -> ( ( ( R u. ( dom R X. ran R ) ) o. R ) u. ( ( R u. ( dom R X. ran R ) ) o. ( dom R X. ran R ) ) ) = ( ( R u. ( dom R X. ran R ) ) o. ( dom R X. ran R ) ) )
26 coundir 5637 . . . . . 6 |- ( ( R u. ( dom R X. ran R ) ) o. ( dom R X. ran R ) ) = ( ( R o. ( dom R X. ran R ) ) u. ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) )
27 coss1 5277 . . . . . . . . . 10 |- ( `' `' R C_ `' ( ran R X. dom R ) -> ( `' `' R o. ( dom R X. ran R ) ) C_ ( `' ( ran R X. dom R ) o. ( dom R X. ran R ) ) )
2817, 27syl 17 . . . . . . . . 9 |- ( `' R C_ ( ran R X. dom R ) -> ( `' `' R o. ( dom R X. ran R ) ) C_ ( `' ( ran R X. dom R ) o. ( dom R X. ran R ) ) )
29 cocnvcnv1 5646 . . . . . . . . 9 |- ( `' `' R o. ( dom R X. ran R ) ) = ( R o. ( dom R X. ran R ) )
3021coeq1i 5281 . . . . . . . . 9 |- ( `' ( ran R X. dom R ) o. ( dom R X. ran R ) ) = ( ( dom R X. ran R ) o. ( dom R X. ran R ) )
3128, 29, 303sstr3g 3645 . . . . . . . 8 |- ( `' R C_ ( ran R X. dom R ) -> ( R o. ( dom R X. ran R ) ) C_ ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) )
32 ssequn1 3783 . . . . . . . 8 |- ( ( R o. ( dom R X. ran R ) ) C_ ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) <-> ( ( R o. ( dom R X. ran R ) ) u. ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) ) = ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) )
3331, 32sylib 208 . . . . . . 7 |- ( `' R C_ ( ran R X. dom R ) -> ( ( R o. ( dom R X. ran R ) ) u. ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) ) = ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) )
34 xptrrel 13719 . . . . . . . . 9 |- ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) C_ ( dom R X. ran R )
35 ssun2 3777 . . . . . . . . 9 |- ( dom R X. ran R ) C_ ( R u. ( dom R X. ran R ) )
3634, 35sstri 3612 . . . . . . . 8 |- ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) C_ ( R u. ( dom R X. ran R ) )
3736a1i 11 . . . . . . 7 |- ( `' R C_ ( ran R X. dom R ) -> ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) C_ ( R u. ( dom R X. ran R ) ) )
3833, 37eqsstrd 3639 . . . . . 6 |- ( `' R C_ ( ran R X. dom R ) -> ( ( R o. ( dom R X. ran R ) ) u. ( ( dom R X. ran R ) o. ( dom R X. ran R ) ) ) C_ ( R u. ( dom R X. ran R ) ) )
3926, 38syl5eqss 3649 . . . . 5 |- ( `' R C_ ( ran R X. dom R ) -> ( ( R u. ( dom R X. ran R ) ) o. ( dom R X. ran R ) ) C_ ( R u. ( dom R X. ran R ) ) )
4025, 39eqsstrd 3639 . . . 4 |- ( `' R C_ ( ran R X. dom R ) -> ( ( ( R u. ( dom R X. ran R ) ) o. R ) u. ( ( R u. ( dom R X. ran R ) ) o. ( dom R X. ran R ) ) ) C_ ( R u. ( dom R X. ran R ) ) )
4116, 40syl5eqss 3649 . . 3 |- ( `' R C_ ( ran R X. dom R ) -> ( ( R u. ( dom R X. ran R ) ) o. ( R u. ( dom R X. ran R ) ) ) C_ ( R u. ( dom R X. ran R ) ) )
4215, 41mp1i 13 . 2 |- ( ph -> ( ( R u. ( dom R X. ran R ) ) o. ( R u. ( dom R X. ran R ) ) ) C_ ( R u. ( dom R X. ran R ) ) )
439, 12, 14, 42clublem 37917 1 |- ( ph -> |^| { x | ( R C_ x /\ ( x o. x ) C_ x ) } C_ ( R u. ( dom R X. ran R ) ) )
Colors of variables: wff setvar class
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   `'ccnv 5113   dom cdm 5114   ran crn 5115   o. ccom 5118
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:  trclubNEW  37926
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