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Theorem trrelsuperrel2dg 36123
Description: Concrete construction of a superclass of relation  R which is a transitive relation. (Contributed by RP, 20-Jul-2020.)
Hypothesis
Ref Expression
trrelsuperrel2dg.s  |-  ( ph  ->  S  =  ( R  u.  ( dom  R  X.  ran  R ) ) )
Assertion
Ref Expression
trrelsuperrel2dg  |-  ( ph  ->  ( R  C_  S  /\  ( S  o.  S
)  C_  S )
)

Proof of Theorem trrelsuperrel2dg
StepHypRef Expression
1 ssun1 3629 . . 3  |-  R  C_  ( R  u.  ( dom  R  X.  ran  R
) )
2 trrelsuperrel2dg.s . . 3  |-  ( ph  ->  S  =  ( R  u.  ( dom  R  X.  ran  R ) ) )
31, 2syl5sseqr 3513 . 2  |-  ( ph  ->  R  C_  S )
4 xptrrel 13033 . . . . 5  |-  ( ( dom  R  X.  ran  R )  o.  ( dom 
R  X.  ran  R
) )  C_  ( dom  R  X.  ran  R
)
5 ssun2 3630 . . . . 5  |-  ( dom 
R  X.  ran  R
)  C_  ( R  u.  ( dom  R  X.  ran  R ) )
64, 5sstri 3473 . . . 4  |-  ( ( dom  R  X.  ran  R )  o.  ( dom 
R  X.  ran  R
) )  C_  ( R  u.  ( dom  R  X.  ran  R ) )
76a1i 11 . . 3  |-  ( ph  ->  ( ( dom  R  X.  ran  R )  o.  ( dom  R  X.  ran  R ) )  C_  ( R  u.  ( dom  R  X.  ran  R
) ) )
82, 2coeq12d 5015 . . . 4  |-  ( ph  ->  ( S  o.  S
)  =  ( ( R  u.  ( dom 
R  X.  ran  R
) )  o.  ( R  u.  ( dom  R  X.  ran  R ) ) ) )
9 coundir 5353 . . . . . 6  |-  ( ( R  u.  ( dom 
R  X.  ran  R
) )  o.  ( R  u.  ( dom  R  X.  ran  R ) ) )  =  ( ( R  o.  ( R  u.  ( dom  R  X.  ran  R ) ) )  u.  (
( dom  R  X.  ran  R )  o.  ( R  u.  ( dom  R  X.  ran  R ) ) ) )
10 relcnv 5223 . . . . . . 7  |-  Rel  `' `' R
11 cocnvcnv1 5362 . . . . . . . . 9  |-  ( `' `' R  o.  ( R  u.  ( dom  R  X.  ran  R ) ) )  =  ( R  o.  ( R  u.  ( dom  R  X.  ran  R ) ) )
12 relssdmrn 5372 . . . . . . . . . . 11  |-  ( Rel  `' `' R  ->  `' `' R  C_  ( dom  `' `' R  X.  ran  `' `' R ) )
13 dmcnvcnv 5073 . . . . . . . . . . . 12  |-  dom  `' `' R  =  dom  R
14 rncnvcnv 5074 . . . . . . . . . . . 12  |-  ran  `' `' R  =  ran  R
1513, 14xpeq12i 4872 . . . . . . . . . . 11  |-  ( dom  `' `' R  X.  ran  `' `' R )  =  ( dom  R  X.  ran  R )
1612, 15syl6sseq 3510 . . . . . . . . . 10  |-  ( Rel  `' `' R  ->  `' `' R  C_  ( dom  R  X.  ran  R ) )
17 coss1 5006 . . . . . . . . . 10  |-  ( `' `' R  C_  ( dom 
R  X.  ran  R
)  ->  ( `' `' R  o.  ( R  u.  ( dom  R  X.  ran  R ) ) )  C_  (
( dom  R  X.  ran  R )  o.  ( R  u.  ( dom  R  X.  ran  R ) ) ) )
1816, 17syl 17 . . . . . . . . 9  |-  ( Rel  `' `' R  ->  ( `' `' R  o.  ( R  u.  ( dom  R  X.  ran  R ) ) )  C_  (
( dom  R  X.  ran  R )  o.  ( R  u.  ( dom  R  X.  ran  R ) ) ) )
1911, 18syl5eqssr 3509 . . . . . . . 8  |-  ( Rel  `' `' R  ->  ( R  o.  ( R  u.  ( dom  R  X.  ran  R ) ) )  C_  ( ( dom  R  X.  ran  R )  o.  ( R  u.  ( dom  R  X.  ran  R
) ) ) )
20 ssequn1 3636 . . . . . . . 8  |-  ( ( R  o.  ( R  u.  ( dom  R  X.  ran  R ) ) )  C_  ( ( dom  R  X.  ran  R
)  o.  ( R  u.  ( dom  R  X.  ran  R ) ) )  <->  ( ( R  o.  ( R  u.  ( dom  R  X.  ran  R ) ) )  u.  ( ( dom  R  X.  ran  R )  o.  ( R  u.  ( dom  R  X.  ran  R
) ) ) )  =  ( ( dom 
R  X.  ran  R
)  o.  ( R  u.  ( dom  R  X.  ran  R ) ) ) )
2119, 20sylib 199 . . . . . . 7  |-  ( Rel  `' `' R  ->  ( ( R  o.  ( R  u.  ( dom  R  X.  ran  R ) ) )  u.  ( ( dom  R  X.  ran  R )  o.  ( R  u.  ( dom  R  X.  ran  R ) ) ) )  =  ( ( dom  R  X.  ran  R )  o.  ( R  u.  ( dom  R  X.  ran  R ) ) ) )
2210, 21ax-mp 5 . . . . . 6  |-  ( ( R  o.  ( R  u.  ( dom  R  X.  ran  R ) ) )  u.  ( ( dom  R  X.  ran  R )  o.  ( R  u.  ( dom  R  X.  ran  R ) ) ) )  =  ( ( dom  R  X.  ran  R )  o.  ( R  u.  ( dom  R  X.  ran  R ) ) )
239, 22eqtri 2451 . . . . 5  |-  ( ( R  u.  ( dom 
R  X.  ran  R
) )  o.  ( R  u.  ( dom  R  X.  ran  R ) ) )  =  ( ( dom  R  X.  ran  R )  o.  ( R  u.  ( dom  R  X.  ran  R ) ) )
24 coundi 5352 . . . . . 6  |-  ( ( dom  R  X.  ran  R )  o.  ( R  u.  ( dom  R  X.  ran  R ) ) )  =  ( ( ( dom  R  X.  ran  R )  o.  R
)  u.  ( ( dom  R  X.  ran  R )  o.  ( dom 
R  X.  ran  R
) ) )
25 cocnvcnv2 5363 . . . . . . . . 9  |-  ( ( dom  R  X.  ran  R )  o.  `' `' R )  =  ( ( dom  R  X.  ran  R )  o.  R
)
26 coss2 5007 . . . . . . . . . 10  |-  ( `' `' R  C_  ( dom 
R  X.  ran  R
)  ->  ( ( dom  R  X.  ran  R
)  o.  `' `' R )  C_  (
( dom  R  X.  ran  R )  o.  ( dom  R  X.  ran  R
) ) )
2716, 26syl 17 . . . . . . . . 9  |-  ( Rel  `' `' R  ->  ( ( dom  R  X.  ran  R )  o.  `' `' R )  C_  (
( dom  R  X.  ran  R )  o.  ( dom  R  X.  ran  R
) ) )
2825, 27syl5eqssr 3509 . . . . . . . 8  |-  ( Rel  `' `' R  ->  ( ( dom  R  X.  ran  R )  o.  R ) 
C_  ( ( dom 
R  X.  ran  R
)  o.  ( dom 
R  X.  ran  R
) ) )
29 ssequn1 3636 . . . . . . . 8  |-  ( ( ( dom  R  X.  ran  R )  o.  R
)  C_  ( ( dom  R  X.  ran  R
)  o.  ( dom 
R  X.  ran  R
) )  <->  ( (
( dom  R  X.  ran  R )  o.  R
)  u.  ( ( dom  R  X.  ran  R )  o.  ( dom 
R  X.  ran  R
) ) )  =  ( ( dom  R  X.  ran  R )  o.  ( dom  R  X.  ran  R ) ) )
3028, 29sylib 199 . . . . . . 7  |-  ( Rel  `' `' R  ->  ( ( ( dom  R  X.  ran  R )  o.  R
)  u.  ( ( dom  R  X.  ran  R )  o.  ( dom 
R  X.  ran  R
) ) )  =  ( ( dom  R  X.  ran  R )  o.  ( dom  R  X.  ran  R ) ) )
3110, 30ax-mp 5 . . . . . 6  |-  ( ( ( dom  R  X.  ran  R )  o.  R
)  u.  ( ( dom  R  X.  ran  R )  o.  ( dom 
R  X.  ran  R
) ) )  =  ( ( dom  R  X.  ran  R )  o.  ( dom  R  X.  ran  R ) )
3224, 31eqtri 2451 . . . . 5  |-  ( ( dom  R  X.  ran  R )  o.  ( R  u.  ( dom  R  X.  ran  R ) ) )  =  ( ( dom  R  X.  ran  R )  o.  ( dom 
R  X.  ran  R
) )
3323, 32eqtri 2451 . . . 4  |-  ( ( R  u.  ( dom 
R  X.  ran  R
) )  o.  ( R  u.  ( dom  R  X.  ran  R ) ) )  =  ( ( dom  R  X.  ran  R )  o.  ( dom  R  X.  ran  R
) )
348, 33syl6eq 2479 . . 3  |-  ( ph  ->  ( S  o.  S
)  =  ( ( dom  R  X.  ran  R )  o.  ( dom 
R  X.  ran  R
) ) )
357, 34, 23sstr4d 3507 . 2  |-  ( ph  ->  ( S  o.  S
)  C_  S )
363, 35jca 534 1  |-  ( ph  ->  ( R  C_  S  /\  ( S  o.  S
)  C_  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    u. cun 3434    C_ wss 3436    X. cxp 4848   `'ccnv 4849   dom cdm 4850   ran crn 4851    o. ccom 4854   Rel wrel 4855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-opab 4480  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862
This theorem is referenced by: (None)
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