Theorem brfvrcld2 37984

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

Step	Hyp	Ref	Expression
1		brfvrcld2.r	. . 3 |- ( ph -> R e. _V )
2	1	brfvrcld 37983	. 2 |- ( ph -> ( A ( r* ` R ) B <-> ( A ( R ^r 0 ) B \/ A ( R ^r 1 ) B ) ) )
3		relexp0g 13762	. . . . . 6 |- ( R e. _V -> ( R ^r 0 ) = ( _I |` ( dom R u. ran R ) ) )
4	1, 3	syl 17	. . . . 5 |- ( ph -> ( R ^r 0 ) = ( _I |` ( dom R u. ran R ) ) )
5	4	breqd 4664	. . . 4 |- ( ph -> ( A ( R ^r 0 ) B <-> A ( _I |` ( dom R u. ran R ) ) B ) )
6		relres 5426	. . . . . . . 8 |- Rel ( _I |` ( dom R u. ran R ) )
7	6	releldmi 5362	. . . . . . 7 |- ( A ( _I |` ( dom R u. ran R ) ) B -> A e. dom ( _I |` ( dom R u. ran R ) ) )
8	6	relelrni 5363	. . . . . . 7 |- ( A ( _I |` ( dom R u. ran R ) ) B -> B e. ran ( _I |` ( dom R u. ran R ) ) )
9		dmresi 5457	. . . . . . . . . 10 |- dom ( _I |` ( dom R u. ran R ) ) = ( dom R u. ran R )
10	9	eleq2i 2693	. . . . . . . . 9 |- ( A e. dom ( _I |` ( dom R u. ran R ) ) <-> A e. ( dom R u. ran R ) )
11	10	biimpi 206	. . . . . . . 8 |- ( A e. dom ( _I |` ( dom R u. ran R ) ) -> A e. ( dom R u. ran R ) )
12		rnresi 5479	. . . . . . . . . 10 |- ran ( _I |` ( dom R u. ran R ) ) = ( dom R u. ran R )
13	12	eleq2i 2693	. . . . . . . . 9 |- ( B e. ran ( _I |` ( dom R u. ran R ) ) <-> B e. ( dom R u. ran R ) )
14	13	biimpi 206	. . . . . . . 8 |- ( B e. ran ( _I |` ( dom R u. ran R ) ) -> B e. ( dom R u. ran R ) )
15	11, 14	anim12i 590	. . . . . . 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 ) ) )
16	7, 8, 15	syl2anc 693	. . . . . 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 5407	. . . . . 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 ) )
18	16, 17	biadan2 674	. . . . 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 1039	. . . . 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 ) )
20	18, 19	bitr4i 267	. . . 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 ) )
21	5, 20	syl6bb 276	. . 3 |- ( ph -> ( A ( R ^r 0 ) B <-> ( A e. ( dom R u. ran R ) /\ B e. ( dom R u. ran R ) /\ A = B ) ) )
22	1	relexp1d 13771	. . . 4 |- ( ph -> ( R ^r 1 ) = R )
23	22	breqd 4664	. . 3 |- ( ph -> ( A ( R ^r 1 ) B <-> A R B ) )
24	21, 23	orbi12d 746	. 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 ) ) )
25	2, 24	bitrd 268	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 ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   class class class wbr 4653   _I cid 5023   dom cdm 5114   ran crn 5115   |` cres 5116   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   ^r crelexp 13760   r*crcl 37964

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-relexp 13761  df-rcl 37965

This theorem is referenced by: (None)