Theorem brfvrcld2 43122

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 43121	. 2 ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ (𝐴(𝑅↑𝑟0)𝐵 ∨ 𝐴(𝑅↑𝑟1)𝐵)))
3		relexp0g 15002	. . . . . 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 5159	. . . 4 ⊢ (𝜑 → (𝐴(𝑅↑𝑟0)𝐵 ↔ 𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵))
6		relres 6014	. . . . . . . 8 ⊢ Rel ( I ↾ (dom 𝑅 ∪ ran 𝑅))
7	6	releldmi 5950	. . . . . . 7 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 → 𝐴 ∈ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
8	6	relelrni 5951	. . . . . . 7 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 → 𝐵 ∈ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
9		dmresi 6055	. . . . . . . . . 10 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
10	9	eleq2i 2821	. . . . . . . . 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 6078	. . . . . . . . . 10 ⊢ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
13	12	eleq2i 2821	. . . . . . . . 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 5996	. . . . . 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 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 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 15009	. . . 4 ⊢ (𝜑 → (𝑅↑𝑟1) = 𝑅)
23	22	breqd 5159	. . 3 ⊢ (𝜑 → (𝐴(𝑅↑𝑟1)𝐵 ↔ 𝐴𝑅𝐵))
24	21, 23	orbi12d 917	. 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 205   ∧ wa 395   ∨ wo 846   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  Vcvv 3471   ∪ cun 3945   class class class wbr 5148   I cid 5575  dom cdm 5678  ran crn 5679   ↾ cres 5680  ‘cfv 6548  (class class class)co 7420  0cc0 11139  1c1 11140  ↑_𝑟crelexp 14999  r*crcl 43102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-n0 12504  df-z 12590  df-uz 12854  df-seq 14000  df-relexp 15000  df-rcl 43103

This theorem is referenced by: (None)

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