Theorem brfvrcld2 43674

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 43673	. 2 ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ (𝐴(𝑅↑𝑟0)𝐵 ∨ 𝐴(𝑅↑𝑟1)𝐵)))
3		relexp0g 14964	. . . . . 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 5113	. . . 4 ⊢ (𝜑 → (𝐴(𝑅↑𝑟0)𝐵 ↔ 𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵))
6		relres 5965	. . . . . . . 8 ⊢ Rel ( I ↾ (dom 𝑅 ∪ ran 𝑅))
7	6	releldmi 5901	. . . . . . 7 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 → 𝐴 ∈ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
8	6	relelrni 5902	. . . . . . 7 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 → 𝐵 ∈ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
9		dmresi 6012	. . . . . . . . . 10 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
10	9	eleq2i 2820	. . . . . . . . 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 6035	. . . . . . . . . 10 ⊢ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
13	12	eleq2i 2820	. . . . . . . . 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 613	. . . . . . 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 584	. . . . . 6 ⊢ (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 → (𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅)))
17		resieq 5950	. . . . . 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 1088	. . . . 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 14971	. . . 4 ⊢ (𝜑 → (𝑅↑𝑟1) = 𝑅)
23	22	breqd 5113	. . 3 ⊢ (𝜑 → (𝐴(𝑅↑𝑟1)𝐵 ↔ 𝐴𝑅𝐵))
24	21, 23	orbi12d 918	. 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 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ∪ cun 3909   class class class wbr 5102   I cid 5525  dom cdm 5631  ran crn 5632   ↾ cres 5633  ‘cfv 6499  (class class class)co 7369  0cc0 11044  1c1 11045  ↑_𝑟crelexp 14961  r*crcl 43654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  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 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-n0 12419  df-z 12506  df-uz 12770  df-seq 13943  df-relexp 14962  df-rcl 43655

This theorem is referenced by: (None)