Theorem brfvrcld2 41231

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 41230	. 2 ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ (𝐴(𝑅↑𝑟0)𝐵 ∨ 𝐴(𝑅↑𝑟1)𝐵)))
3		relexp0g 14677	. . . . . 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 5086	. . . 4 ⊢ (𝜑 → (𝐴(𝑅↑𝑟0)𝐵 ↔ 𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵))
6		relres 5914	. . . . . . . 8 ⊢ Rel ( I ↾ (dom 𝑅 ∪ ran 𝑅))
7	6	releldmi 5851	. . . . . . 7 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 → 𝐴 ∈ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
8	6	relelrni 5852	. . . . . . 7 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 → 𝐵 ∈ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
9		dmresi 5955	. . . . . . . . . 10 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
10	9	eleq2i 2828	. . . . . . . . 9 ⊢ (𝐴 ∈ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↔ 𝐴 ∈ (dom 𝑅 ∪ ran 𝑅))
11	10	biimpi 215	. . . . . . . 8 ⊢ (𝐴 ∈ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) → 𝐴 ∈ (dom 𝑅 ∪ ran 𝑅))
12		rnresi 5977	. . . . . . . . . 10 ⊢ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
13	12	eleq2i 2828	. . . . . . . . 9 ⊢ (𝐵 ∈ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↔ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅))
14	13	biimpi 215	. . . . . . . 8 ⊢ (𝐵 ∈ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) → 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅))
15	11, 14	anim12i 612	. . . . . . 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 583	. . . . . 6 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 → (𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅)))
17		resieq 5896	. . . . . 6 ⊢ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅)) → (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 ↔ 𝐴 = 𝐵))
18	16, 17	biadanii 818	. . . . 5 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 ↔ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅)) ∧ 𝐴 = 𝐵))
19		df-3an 1087	. . . . 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 277	. . . 4 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 ↔ (𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵))
21	5, 20	bitrdi 286	. . 3 ⊢ (𝜑 → (𝐴(𝑅↑𝑟0)𝐵 ↔ (𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵)))
22	1	relexp1d 14684	. . . 4 ⊢ (𝜑 → (𝑅↑𝑟1) = 𝑅)
23	22	breqd 5086	. . 3 ⊢ (𝜑 → (𝐴(𝑅↑𝑟1)𝐵 ↔ 𝐴𝑅𝐵))
24	21, 23	orbi12d 915	. 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 278	1 ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵) ∨ 𝐴𝑅𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2107  Vcvv 3427   ∪ cun 3886   class class class wbr 5075   I cid 5484  dom cdm 5585  ran crn 5586   ↾ cres 5587  ‘cfv 6423  (class class class)co 7260  0cc0 10818  1c1 10819  ↑_𝑟crelexp 14674  r*crcl 41211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5210  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7571  ax-cnex 10874  ax-resscn 10875  ax-1cn 10876  ax-icn 10877  ax-addcl 10878  ax-addrcl 10879  ax-mulcl 10880  ax-mulrcl 10881  ax-mulcom 10882  ax-addass 10883  ax-mulass 10884  ax-distr 10885  ax-i2m1 10886  ax-1ne0 10887  ax-1rid 10888  ax-rnegex 10889  ax-rrecex 10890  ax-cnre 10891  ax-pre-lttri 10892  ax-pre-lttrn 10893  ax-pre-ltadd 10894  ax-pre-mulgt0 10895

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3067  df-rex 3068  df-reu 3069  df-rab 3071  df-v 3429  df-sbc 3717  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4842  df-int 4882  df-iun 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5485  df-eprel 5491  df-po 5499  df-so 5500  df-fr 5540  df-we 5542  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-pred 6196  df-ord 6259  df-on 6260  df-lim 6261  df-suc 6262  df-iota 6381  df-fun 6425  df-fn 6426  df-f 6427  df-f1 6428  df-fo 6429  df-f1o 6430  df-fv 6431  df-riota 7217  df-ov 7263  df-oprab 7264  df-mpo 7265  df-om 7693  df-2nd 7810  df-frecs 8073  df-wrecs 8104  df-recs 8178  df-rdg 8217  df-er 8461  df-en 8697  df-dom 8698  df-sdom 8699  df-pnf 10958  df-mnf 10959  df-xr 10960  df-ltxr 10961  df-le 10962  df-sub 11153  df-neg 11154  df-nn 11920  df-n0 12180  df-z 12266  df-uz 12528  df-seq 13666  df-relexp 14675  df-rcl 41212

This theorem is referenced by: (None)