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Theorem brfvrcld2 38510
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 (𝜑𝑅 ∈ 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
StepHypRef Expression
1 brfvrcld2.r . . 3 (𝜑𝑅 ∈ V)
21brfvrcld 38509 . 2 (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ (𝐴(𝑅𝑟0)𝐵𝐴(𝑅𝑟1)𝐵)))
3 relexp0g 13970 . . . . . 6 (𝑅 ∈ V → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
41, 3syl 17 . . . . 5 (𝜑 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
54breqd 4797 . . . 4 (𝜑 → (𝐴(𝑅𝑟0)𝐵𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵))
6 relres 5567 . . . . . . . 8 Rel ( I ↾ (dom 𝑅 ∪ ran 𝑅))
76releldmi 5500 . . . . . . 7 (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵𝐴 ∈ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
86relelrni 5501 . . . . . . 7 (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵𝐵 ∈ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
9 dmresi 5598 . . . . . . . . . 10 dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
109eleq2i 2842 . . . . . . . . 9 (𝐴 ∈ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↔ 𝐴 ∈ (dom 𝑅 ∪ ran 𝑅))
1110biimpi 206 . . . . . . . 8 (𝐴 ∈ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) → 𝐴 ∈ (dom 𝑅 ∪ ran 𝑅))
12 rnresi 5620 . . . . . . . . . 10 ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
1312eleq2i 2842 . . . . . . . . 9 (𝐵 ∈ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↔ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅))
1413biimpi 206 . . . . . . . 8 (𝐵 ∈ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) → 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅))
1511, 14anim12i 592 . . . . . . 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)))
167, 8, 15syl2anc 565 . . . . . 6 (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 → (𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅)))
17 resieq 5548 . . . . . 6 ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅)) → (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵𝐴 = 𝐵))
1816, 17biadan2 802 . . . . 5 (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 ↔ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅)) ∧ 𝐴 = 𝐵))
19 df-3an 1073 . . . . 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))
2018, 19bitr4i 267 . . . 4 (𝐴( I ↾ (dom 𝑅 ∪ ran 𝑅))𝐵 ↔ (𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵))
215, 20syl6bb 276 . . 3 (𝜑 → (𝐴(𝑅𝑟0)𝐵 ↔ (𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵)))
221relexp1d 13979 . . . 4 (𝜑 → (𝑅𝑟1) = 𝑅)
2322breqd 4797 . . 3 (𝜑 → (𝐴(𝑅𝑟1)𝐵𝐴𝑅𝐵))
2421, 23orbi12d 892 . 2 (𝜑 → ((𝐴(𝑅𝑟0)𝐵𝐴(𝑅𝑟1)𝐵) ↔ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵) ∨ 𝐴𝑅𝐵)))
252, 24bitrd 268 1 (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵) ∨ 𝐴𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wo 826  w3a 1071   = wceq 1631  wcel 2145  Vcvv 3351  cun 3721   class class class wbr 4786   I cid 5156  dom cdm 5249  ran crn 5250  cres 5251  cfv 6031  (class class class)co 6793  0cc0 10138  1c1 10139  𝑟crelexp 13968  r*crcl 38490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-z 11580  df-uz 11889  df-seq 13009  df-relexp 13969  df-rcl 38491
This theorem is referenced by: (None)
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