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Theorem cnvtrcl0 37933
Description: The converse of the transitive closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
Assertion
Ref Expression
cnvtrcl0 |- ( X e. V -> `' |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } = |^| { y | ( `' X C_ y /\ ( y o. y ) C_ y ) } )
Distinct variable groups:   x, y, V   x, X, y
Proof of Theorem cnvtrcl0
StepHypRef Expression
1 cnvco 5308 . . . . . 6 |- `' ( y o. y ) = ( `' y o. `' y )
2 cnvss 5294 . . . . . 6 |- ( ( y o. y ) C_ y -> `' ( y o. y ) C_ `' y )
31, 2syl5eqssr 3650 . . . . 5 |- ( ( y o. y ) C_ y -> ( `' y o. `' y ) C_ `' y )
4 coundir 5637 . . . . . . 7 |- ( ( `' y u. ( X \ `' `' X ) ) o. ( `' y u. ( X \ `' `' X ) ) ) = ( ( `' y o. ( `' y u. ( X \ `' `' X ) ) ) u. ( ( X \ `' `' X ) o. ( `' y u. ( X \ `' `' X ) ) ) )
5 coundi 5636 . . . . . . . . 9 |- ( `' y o. ( `' y u. ( X \ `' `' X ) ) ) = ( ( `' y o. `' y ) u. ( `' y o. ( X \ `' `' X ) ) )
6 ssid 3624 . . . . . . . . . 10 |- ( `' y o. `' y ) C_ ( `' y o. `' y )
7 cononrel2 37901 . . . . . . . . . . 11 |- ( `' y o. ( X \ `' `' X ) ) = (/)
8 0ss 3972 . . . . . . . . . . 11 |- (/) C_ ( `' y o. `' y )
97, 8eqsstri 3635 . . . . . . . . . 10 |- ( `' y o. ( X \ `' `' X ) ) C_ ( `' y o. `' y )
106, 9unssi 3788 . . . . . . . . 9 |- ( ( `' y o. `' y ) u. ( `' y o. ( X \ `' `' X ) ) ) C_ ( `' y o. `' y )
115, 10eqsstri 3635 . . . . . . . 8 |- ( `' y o. ( `' y u. ( X \ `' `' X ) ) ) C_ ( `' y o. `' y )
12 cononrel1 37900 . . . . . . . . 9 |- ( ( X \ `' `' X ) o. ( `' y u. ( X \ `' `' X ) ) ) = (/)
1312, 8eqsstri 3635 . . . . . . . 8 |- ( ( X \ `' `' X ) o. ( `' y u. ( X \ `' `' X ) ) ) C_ ( `' y o. `' y )
1411, 13unssi 3788 . . . . . . 7 |- ( ( `' y o. ( `' y u. ( X \ `' `' X ) ) ) u. ( ( X \ `' `' X ) o. ( `' y u. ( X \ `' `' X ) ) ) ) C_ ( `' y o. `' y )
154, 14eqsstri 3635 . . . . . 6 |- ( ( `' y u. ( X \ `' `' X ) ) o. ( `' y u. ( X \ `' `' X ) ) ) C_ ( `' y o. `' y )
16 id 22 . . . . . 6 |- ( ( `' y o. `' y ) C_ `' y -> ( `' y o. `' y ) C_ `' y )
1715, 16syl5ss 3614 . . . . 5 |- ( ( `' y o. `' y ) C_ `' y -> ( ( `' y u. ( X \ `' `' X ) ) o. ( `' y u. ( X \ `' `' X ) ) ) C_ `' y )
18 ssun3 3778 . . . . 5 |- ( ( ( `' y u. ( X \ `' `' X ) ) o. ( `' y u. ( X \ `' `' X ) ) ) C_ `' y -> ( ( `' y u. ( X \ `' `' X ) ) o. ( `' y u. ( X \ `' `' X ) ) ) C_ ( `' y u. ( X \ `' `' X ) ) )
193, 17, 183syl 18 . . . 4 |- ( ( y o. y ) C_ y -> ( ( `' y u. ( X \ `' `' X ) ) o. ( `' y u. ( X \ `' `' X ) ) ) C_ ( `' y u. ( X \ `' `' X ) ) )
20 id 22 . . . . . 6 |- ( x = ( `' y u. ( X \ `' `' X ) ) -> x = ( `' y u. ( X \ `' `' X ) ) )
2120, 20coeq12d 5286 . . . . 5 |- ( x = ( `' y u. ( X \ `' `' X ) ) -> ( x o. x ) = ( ( `' y u. ( X \ `' `' X ) ) o. ( `' y u. ( X \ `' `' X ) ) ) )
2221, 20sseq12d 3634 . . . 4 |- ( x = ( `' y u. ( X \ `' `' X ) ) -> ( ( x o. x ) C_ x <-> ( ( `' y u. ( X \ `' `' X ) ) o. ( `' y u. ( X \ `' `' X ) ) ) C_ ( `' y u. ( X \ `' `' X ) ) ) )
2319, 22syl5ibr 236 . . 3 |- ( x = ( `' y u. ( X \ `' `' X ) ) -> ( ( y o. y ) C_ y -> ( x o. x ) C_ x ) )
2423adantl 482 . 2 |- ( ( X e. V /\ x = ( `' y u. ( X \ `' `' X ) ) ) -> ( ( y o. y ) C_ y -> ( x o. x ) C_ x ) )
25 cnvco 5308 . . . . 5 |- `' ( x o. x ) = ( `' x o. `' x )
26 cnvss 5294 . . . . 5 |- ( ( x o. x ) C_ x -> `' ( x o. x ) C_ `' x )
2725, 26syl5eqssr 3650 . . . 4 |- ( ( x o. x ) C_ x -> ( `' x o. `' x ) C_ `' x )
28 id 22 . . . . . 6 |- ( y = `' x -> y = `' x )
2928, 28coeq12d 5286 . . . . 5 |- ( y = `' x -> ( y o. y ) = ( `' x o. `' x ) )
3029, 28sseq12d 3634 . . . 4 |- ( y = `' x -> ( ( y o. y ) C_ y <-> ( `' x o. `' x ) C_ `' x ) )
3127, 30syl5ibr 236 . . 3 |- ( y = `' x -> ( ( x o. x ) C_ x -> ( y o. y ) C_ y ) )
3231adantl 482 . 2 |- ( ( X e. V /\ y = `' x ) -> ( ( x o. x ) C_ x -> ( y o. y ) C_ y ) )
33 id 22 . . . 4 |- ( x = ( X u. ( dom X X. ran X ) ) -> x = ( X u. ( dom X X. ran X ) ) )
3433, 33coeq12d 5286 . . 3 |- ( x = ( X u. ( dom X X. ran X ) ) -> ( x o. x ) = ( ( X u. ( dom X X. ran X ) ) o. ( X u. ( dom X X. ran X ) ) ) )
3534, 33sseq12d 3634 . 2 |- ( x = ( X u. ( dom X X. ran X ) ) -> ( ( x o. x ) C_ x <-> ( ( X u. ( dom X X. ran X ) ) o. ( X u. ( dom X X. ran X ) ) ) C_ ( X u. ( dom X X. ran X ) ) ) )
36 ssun1 3776 . . 3 |- X C_ ( X u. ( dom X X. ran X ) )
3736a1i 11 . 2 |- ( X e. V -> X C_ ( X u. ( dom X X. ran X ) ) )
38 trclexlem 13733 . 2 |- ( X e. V -> ( X u. ( dom X X. ran X ) ) e. _V )
39 coundir 5637 . . . . 5 |- ( ( X u. ( dom X X. ran X ) ) o. ( X u. ( dom X X. ran X ) ) ) = ( ( X o. ( X u. ( dom X X. ran X ) ) ) u. ( ( dom X X. ran X ) o. ( X u. ( dom X X. ran X ) ) ) )
40 coundi 5636 . . . . . . 7 |- ( X o. ( X u. ( dom X X. ran X ) ) ) = ( ( X o. X ) u. ( X o. ( dom X X. ran X ) ) )
41 cossxp 5658 . . . . . . . 8 |- ( X o. X ) C_ ( dom X X. ran X )
42 cossxp 5658 . . . . . . . . 9 |- ( X o. ( dom X X. ran X ) ) C_ ( dom ( dom X X. ran X ) X. ran X )
43 dmxpss 5565 . . . . . . . . . 10 |- dom ( dom X X. ran X ) C_ dom X
44 xpss1 5228 . . . . . . . . . 10 |- ( dom ( dom X X. ran X ) C_ dom X -> ( dom ( dom X X. ran X ) X. ran X ) C_ ( dom X X. ran X ) )
4543, 44ax-mp 5 . . . . . . . . 9 |- ( dom ( dom X X. ran X ) X. ran X ) C_ ( dom X X. ran X )
4642, 45sstri 3612 . . . . . . . 8 |- ( X o. ( dom X X. ran X ) ) C_ ( dom X X. ran X )
4741, 46unssi 3788 . . . . . . 7 |- ( ( X o. X ) u. ( X o. ( dom X X. ran X ) ) ) C_ ( dom X X. ran X )
4840, 47eqsstri 3635 . . . . . 6 |- ( X o. ( X u. ( dom X X. ran X ) ) ) C_ ( dom X X. ran X )
49 coundi 5636 . . . . . . 7 |- ( ( dom X X. ran X ) o. ( X u. ( dom X X. ran X ) ) ) = ( ( ( dom X X. ran X ) o. X ) u. ( ( dom X X. ran X ) o. ( dom X X. ran X ) ) )
50 cossxp 5658 . . . . . . . . 9 |- ( ( dom X X. ran X ) o. X ) C_ ( dom X X. ran ( dom X X. ran X ) )
51 rnxpss 5566 . . . . . . . . . 10 |- ran ( dom X X. ran X ) C_ ran X
52 xpss2 5229 . . . . . . . . . 10 |- ( ran ( dom X X. ran X ) C_ ran X -> ( dom X X. ran ( dom X X. ran X ) ) C_ ( dom X X. ran X ) )
5351, 52ax-mp 5 . . . . . . . . 9 |- ( dom X X. ran ( dom X X. ran X ) ) C_ ( dom X X. ran X )
5450, 53sstri 3612 . . . . . . . 8 |- ( ( dom X X. ran X ) o. X ) C_ ( dom X X. ran X )
55 xptrrel 13719 . . . . . . . 8 |- ( ( dom X X. ran X ) o. ( dom X X. ran X ) ) C_ ( dom X X. ran X )
5654, 55unssi 3788 . . . . . . 7 |- ( ( ( dom X X. ran X ) o. X ) u. ( ( dom X X. ran X ) o. ( dom X X. ran X ) ) ) C_ ( dom X X. ran X )
5749, 56eqsstri 3635 . . . . . 6 |- ( ( dom X X. ran X ) o. ( X u. ( dom X X. ran X ) ) ) C_ ( dom X X. ran X )
5848, 57unssi 3788 . . . . 5 |- ( ( X o. ( X u. ( dom X X. ran X ) ) ) u. ( ( dom X X. ran X ) o. ( X u. ( dom X X. ran X ) ) ) ) C_ ( dom X X. ran X )
5939, 58eqsstri 3635 . . . 4 |- ( ( X u. ( dom X X. ran X ) ) o. ( X u. ( dom X X. ran X ) ) ) C_ ( dom X X. ran X )
60 ssun2 3777 . . . 4 |- ( dom X X. ran X ) C_ ( X u. ( dom X X. ran X ) )
6159, 60sstri 3612 . . 3 |- ( ( X u. ( dom X X. ran X ) ) o. ( X u. ( dom X X. ran X ) ) ) C_ ( X u. ( dom X X. ran X ) )
6261a1i 11 . 2 |- ( X e. V -> ( ( X u. ( dom X X. ran X ) ) o. ( X u. ( dom X X. ran X ) ) ) C_ ( X u. ( dom X X. ran X ) ) )
6324, 32, 35, 37, 38, 62clcnvlem 37930 1 |- ( X e. V -> `' |^| { x | ( X C_ x /\ ( x o. x ) C_ x ) } = |^| { y | ( `' X C_ y /\ ( y o. y ) C_ y ) } )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   |^|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-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-1st 7168  df-2nd 7169
This theorem is referenced by: (None)
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