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Theorem brfvrcld2 36254
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 36253 . 2  |-  ( ph  ->  ( A ( r* `  R ) B  <->  ( A ( R ^r  0 ) B  \/  A
( R ^r 
1 ) B ) ) )
3 relexp0g 13085 . . . . . 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 4434 . . . 4  |-  ( ph  ->  ( A ( R ^r  0 ) B  <->  A (  _I  |`  ( dom  R  u.  ran  R
) ) B ) )
6 relres 5151 . . . . . . . 8  |-  Rel  (  _I  |`  ( dom  R  u.  ran  R ) )
76releldmi 5090 . . . . . . 7  |-  ( A (  _I  |`  ( dom  R  u.  ran  R
) ) B  ->  A  e.  dom  (  _I  |`  ( dom  R  u.  ran  R ) ) )
86relelrni 5091 . . . . . . 7  |-  ( A (  _I  |`  ( dom  R  u.  ran  R
) ) B  ->  B  e.  ran  (  _I  |`  ( dom  R  u.  ran  R ) ) )
9 dmresi 5179 . . . . . . . . . 10  |-  dom  (  _I  |`  ( dom  R  u.  ran  R ) )  =  ( dom  R  u.  ran  R )
109eleq2i 2499 . . . . . . . . 9  |-  ( A  e.  dom  (  _I  |`  ( dom  R  u.  ran  R ) )  <->  A  e.  ( dom  R  u.  ran  R ) )
1110biimpi 197 . . . . . . . 8  |-  ( A  e.  dom  (  _I  |`  ( dom  R  u.  ran  R ) )  ->  A  e.  ( dom  R  u.  ran  R ) )
12 rnresi 5200 . . . . . . . . . 10  |-  ran  (  _I  |`  ( dom  R  u.  ran  R ) )  =  ( dom  R  u.  ran  R )
1312eleq2i 2499 . . . . . . . . 9  |-  ( B  e.  ran  (  _I  |`  ( dom  R  u.  ran  R ) )  <->  B  e.  ( dom  R  u.  ran  R ) )
1413biimpi 197 . . . . . . . 8  |-  ( B  e.  ran  (  _I  |`  ( dom  R  u.  ran  R ) )  ->  B  e.  ( dom  R  u.  ran  R ) )
1511, 14anim12i 568 . . . . . . 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 665 . . . . . 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 5134 . . . . . 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 646 . . . . 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 984 . . . . 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 255 . . . 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 264 . . 3  |-  ( ph  ->  ( A ( R ^r  0 ) B  <->  ( A  e.  ( dom  R  u.  ran  R )  /\  B  e.  ( dom  R  u.  ran  R )  /\  A  =  B ) ) )
221relexp1d 13094 . . . 4  |-  ( ph  ->  ( R ^r 
1 )  =  R )
2322breqd 4434 . . 3  |-  ( ph  ->  ( A ( R ^r  1 ) B  <->  A R B ) )
2421, 23orbi12d 714 . 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 256 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 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   _Vcvv 3080    u. cun 3434   class class class wbr 4423    _I cid 4763   dom cdm 4853   ran crn 4854    |` cres 4855   ` cfv 5601  (class class class)co 6305   0cc0 9546   1c1 9547   ^r crelexp 13083   r*crcl 36234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-n0 10877  df-z 10945  df-uz 11167  df-seq 12220  df-relexp 13084  df-rcl 36235
This theorem is referenced by: (None)
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