Theorem brfvrcld2 44119

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 44118	. 2 ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ (𝐴(𝑅↑𝑟0)𝐵 ∨ 𝐴(𝑅↑𝑟1)𝐵)))
3		relexp0g 14984	. . . . . 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 5096	. . . 4 ⊢ (𝜑 → (𝐴(𝑅↑𝑟0)𝐵 ↔ 𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵))
6		relres 5970	. . . . . . . 8 ⊢ Rel ( I ↾ (dom 𝑅 ∪ ran 𝑅))
7	6	releldmi 5903	. . . . . . 7 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 → 𝐴 ∈ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
8	6	relelrni 5904	. . . . . . 7 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 → 𝐵 ∈ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
9		dmresi 6017	. . . . . . . . . 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 216	. . . . . . . 8 ⊢ (𝐴 ∈ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) → 𝐴 ∈ (dom 𝑅 ∪ ran 𝑅))
12		rnresi 6040	. . . . . . . . . 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 216	. . . . . . . 8 ⊢ (𝐵 ∈ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) → 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅))
15	11, 14	anim12i 614	. . . . . . 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 585	. . . . . 6 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 → (𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅)))
17		resieq 5955	. . . . . 6 ⊢ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅)) → (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 ↔ 𝐴 = 𝐵))
18	16, 17	biadanii 822	. . . . 5 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 ↔ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅)) ∧ 𝐴 = 𝐵))
19		df-3an 1089	. . . . 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 278	. . . 4 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 ↔ (𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵))
21	5, 20	bitrdi 287	. . 3 ⊢ (𝜑 → (𝐴(𝑅↑𝑟0)𝐵 ↔ (𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵)))
22	1	relexp1d 14991	. . . 4 ⊢ (𝜑 → (𝑅↑𝑟1) = 𝑅)
23	22	breqd 5096	. . 3 ⊢ (𝜑 → (𝐴(𝑅↑𝑟1)𝐵 ↔ 𝐴𝑅𝐵))
24	21, 23	orbi12d 919	. 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 279	1 ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵) ∨ 𝐴𝑅𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∪ cun 3887   class class class wbr 5085   I cid 5525  dom cdm 5631  ran crn 5632   ↾ cres 5633  ‘cfv 6498  (class class class)co 7367  0cc0 11038  1c1 11039  ↑_𝑟crelexp 14981  r*crcl 44099

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-n0 12438  df-z 12525  df-uz 12789  df-seq 13964  df-relexp 14982  df-rcl 44100

This theorem is referenced by: (None)