Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brfvrcld2 Structured version   Visualization version   Unicode version

Theorem brfvrcld2 36355
Description: If two elements are connected by the reflexive closure of a relation, then they are equal or related by relation. (Contributed by RP, 21-Jul-2020.)
Hypothesis
Ref Expression
brfvrcld2.r  |-  ( ph  ->  R  e.  _V )
Assertion
Ref Expression
brfvrcld2  |-  ( ph  ->  ( A ( r* `  R ) B  <->  ( ( A  e.  ( dom  R  u.  ran  R )  /\  B  e.  ( dom  R  u.  ran  R )  /\  A  =  B )  \/  A R B ) ) )

Proof of Theorem brfvrcld2
StepHypRef Expression
1 brfvrcld2.r . . 3  |-  ( ph  ->  R  e.  _V )
21brfvrcld 36354 . 2  |-  ( ph  ->  ( A ( r* `  R ) B  <->  ( A ( R ^r  0 ) B  \/  A
( R ^r 
1 ) B ) ) )
3 relexp0g 13162 . . . . . 6  |-  ( R  e.  _V  ->  ( R ^r  0 )  =  (  _I  |`  ( dom  R  u.  ran  R
) ) )
41, 3syl 17 . . . . 5  |-  ( ph  ->  ( R ^r 
0 )  =  (  _I  |`  ( dom  R  u.  ran  R ) ) )
54breqd 4406 . . . 4  |-  ( ph  ->  ( A ( R ^r  0 ) B  <->  A (  _I  |`  ( dom  R  u.  ran  R
) ) B ) )
6 relres 5138 . . . . . . . 8  |-  Rel  (  _I  |`  ( dom  R  u.  ran  R ) )
76releldmi 5077 . . . . . . 7  |-  ( A (  _I  |`  ( dom  R  u.  ran  R
) ) B  ->  A  e.  dom  (  _I  |`  ( dom  R  u.  ran  R ) ) )
86relelrni 5078 . . . . . . 7  |-  ( A (  _I  |`  ( dom  R  u.  ran  R
) ) B  ->  B  e.  ran  (  _I  |`  ( dom  R  u.  ran  R ) ) )
9 dmresi 5166 . . . . . . . . . 10  |-  dom  (  _I  |`  ( dom  R  u.  ran  R ) )  =  ( dom  R  u.  ran  R )
109eleq2i 2541 . . . . . . . . 9  |-  ( A  e.  dom  (  _I  |`  ( dom  R  u.  ran  R ) )  <->  A  e.  ( dom  R  u.  ran  R ) )
1110biimpi 199 . . . . . . . 8  |-  ( A  e.  dom  (  _I  |`  ( dom  R  u.  ran  R ) )  ->  A  e.  ( dom  R  u.  ran  R ) )
12 rnresi 5187 . . . . . . . . . 10  |-  ran  (  _I  |`  ( dom  R  u.  ran  R ) )  =  ( dom  R  u.  ran  R )
1312eleq2i 2541 . . . . . . . . 9  |-  ( B  e.  ran  (  _I  |`  ( dom  R  u.  ran  R ) )  <->  B  e.  ( dom  R  u.  ran  R ) )
1413biimpi 199 . . . . . . . 8  |-  ( B  e.  ran  (  _I  |`  ( dom  R  u.  ran  R ) )  ->  B  e.  ( dom  R  u.  ran  R ) )
1511, 14anim12i 576 . . . . . . 7  |-  ( ( A  e.  dom  (  _I  |`  ( dom  R  u.  ran  R ) )  /\  B  e.  ran  (  _I  |`  ( dom 
R  u.  ran  R
) ) )  -> 
( A  e.  ( dom  R  u.  ran  R )  /\  B  e.  ( dom  R  u.  ran  R ) ) )
167, 8, 15syl2anc 673 . . . . . 6  |-  ( A (  _I  |`  ( dom  R  u.  ran  R
) ) B  -> 
( A  e.  ( dom  R  u.  ran  R )  /\  B  e.  ( dom  R  u.  ran  R ) ) )
17 resieq 5121 . . . . . 6  |-  ( ( A  e.  ( dom 
R  u.  ran  R
)  /\  B  e.  ( dom  R  u.  ran  R ) )  ->  ( A (  _I  |`  ( dom  R  u.  ran  R
) ) B  <->  A  =  B ) )
1816, 17biadan2 654 . . . . 5  |-  ( A (  _I  |`  ( dom  R  u.  ran  R
) ) B  <->  ( ( A  e.  ( dom  R  u.  ran  R )  /\  B  e.  ( dom  R  u.  ran  R ) )  /\  A  =  B ) )
19 df-3an 1009 . . . . 5  |-  ( ( A  e.  ( dom 
R  u.  ran  R
)  /\  B  e.  ( dom  R  u.  ran  R )  /\  A  =  B )  <->  ( ( A  e.  ( dom  R  u.  ran  R )  /\  B  e.  ( dom  R  u.  ran  R ) )  /\  A  =  B ) )
2018, 19bitr4i 260 . . . 4  |-  ( A (  _I  |`  ( dom  R  u.  ran  R
) ) B  <->  ( A  e.  ( dom  R  u.  ran  R )  /\  B  e.  ( dom  R  u.  ran  R )  /\  A  =  B ) )
215, 20syl6bb 269 . . 3  |-  ( ph  ->  ( A ( R ^r  0 ) B  <->  ( A  e.  ( dom  R  u.  ran  R )  /\  B  e.  ( dom  R  u.  ran  R )  /\  A  =  B ) ) )
221relexp1d 13171 . . . 4  |-  ( ph  ->  ( R ^r 
1 )  =  R )
2322breqd 4406 . . 3  |-  ( ph  ->  ( A ( R ^r  1 ) B  <->  A R B ) )
2421, 23orbi12d 724 . 2  |-  ( ph  ->  ( ( A ( R ^r  0 ) B  \/  A
( R ^r 
1 ) B )  <-> 
( ( A  e.  ( dom  R  u.  ran  R )  /\  B  e.  ( dom  R  u.  ran  R )  /\  A  =  B )  \/  A R B ) ) )
252, 24bitrd 261 1  |-  ( ph  ->  ( A ( r* `  R ) B  <->  ( ( A  e.  ( dom  R  u.  ran  R )  /\  B  e.  ( dom  R  u.  ran  R )  /\  A  =  B )  \/  A R B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   _Vcvv 3031    u. cun 3388   class class class wbr 4395    _I cid 4749   dom cdm 4839   ran crn 4840    |` cres 4841   ` cfv 5589  (class class class)co 6308   0cc0 9557   1c1 9558   ^r crelexp 13160   r*crcl 36335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-seq 12252  df-relexp 13161  df-rcl 36336
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator