Theorem brfvrcld2 40393

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	⊢ (𝜑 → 𝑅 ∈ V)

Assertion

Ref	Expression
brfvrcld2	⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵) ∨ 𝐴𝑅𝐵)))

Proof of Theorem brfvrcld2

Step	Hyp	Ref	Expression
1		brfvrcld2.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
2	1	brfvrcld 40392	. 2 ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ (𝐴(𝑅↑𝑟0)𝐵 ∨ 𝐴(𝑅↑𝑟1)𝐵)))
3		relexp0g 14373	. . . . . 6 ⊢ (𝑅 ∈ V → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
4	1, 3	syl 17	. . . . 5 ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
5	4	breqd 5041	. . . 4 ⊢ (𝜑 → (𝐴(𝑅↑𝑟0)𝐵 ↔ 𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵))
6		relres 5847	. . . . . . . 8 ⊢ Rel ( I ↾ (dom 𝑅 ∪ ran 𝑅))
7	6	releldmi 5782	. . . . . . 7 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 → 𝐴 ∈ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
8	6	relelrni 5783	. . . . . . 7 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 → 𝐵 ∈ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
9		dmresi 5888	. . . . . . . . . 10 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
10	9	eleq2i 2881	. . . . . . . . 9 ⊢ (𝐴 ∈ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↔ 𝐴 ∈ (dom 𝑅 ∪ ran 𝑅))
11	10	biimpi 219	. . . . . . . 8 ⊢ (𝐴 ∈ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) → 𝐴 ∈ (dom 𝑅 ∪ ran 𝑅))
12		rnresi 5910	. . . . . . . . . 10 ⊢ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
13	12	eleq2i 2881	. . . . . . . . 9 ⊢ (𝐵 ∈ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↔ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅))
14	13	biimpi 219	. . . . . . . 8 ⊢ (𝐵 ∈ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) → 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅))
15	11, 14	anim12i 615	. . . . . . 7 ⊢ ((𝐴 ∈ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∧ 𝐵 ∈ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅))) → (𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅)))
16	7, 8, 15	syl2anc 587	. . . . . 6 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 → (𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅)))
17		resieq 5829	. . . . . 6 ⊢ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅)) → (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 ↔ 𝐴 = 𝐵))
18	16, 17	biadanii 821	. . . . 5 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 ↔ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅)) ∧ 𝐴 = 𝐵))
19		df-3an 1086	. . . . 5 ⊢ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵) ↔ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅)) ∧ 𝐴 = 𝐵))
20	18, 19	bitr4i 281	. . . 4 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 ↔ (𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵))
21	5, 20	syl6bb 290	. . 3 ⊢ (𝜑 → (𝐴(𝑅↑𝑟0)𝐵 ↔ (𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵)))
22	1	relexp1d 14380	. . . 4 ⊢ (𝜑 → (𝑅↑𝑟1) = 𝑅)
23	22	breqd 5041	. . 3 ⊢ (𝜑 → (𝐴(𝑅↑𝑟1)𝐵 ↔ 𝐴𝑅𝐵))
24	21, 23	orbi12d 916	. 2 ⊢ (𝜑 → ((𝐴(𝑅↑𝑟0)𝐵 ∨ 𝐴(𝑅↑𝑟1)𝐵) ↔ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵) ∨ 𝐴𝑅𝐵)))
25	2, 24	bitrd 282	1 ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵) ∨ 𝐴𝑅𝐵)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∪ cun 3879   class class class wbr 5030   I cid 5424  dom cdm 5519  ran crn 5520   ↾ cres 5521  ‘cfv 6324  (class class class)co 7135  0cc0 10526  1c1 10527  ↑_𝑟crelexp 14370  r*crcl 40373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-seq 13365  df-relexp 14371  df-rcl 40374

This theorem is referenced by: (None)