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Mirrors > Home > MPE Home > Th. List > Mathboxes > brfvrcld | Structured version Visualization version GIF version |
Description: If two elements are connected by the reflexive closure of a relation, then they are connected via zero or one instances the relation. (Contributed by RP, 21-Jul-2020.) |
Ref | Expression |
---|---|
brfvrcld.r | ⊢ (𝜑 → 𝑅 ∈ V) |
Ref | Expression |
---|---|
brfvrcld | ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ (𝐴(𝑅↑𝑟0)𝐵 ∨ 𝐴(𝑅↑𝑟1)𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrcl4 37968 | . . 3 ⊢ r* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ {0, 1} (𝑟↑𝑟𝑛)) | |
2 | brfvrcld.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) | |
3 | 0nn0 11307 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
4 | 1nn0 11308 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
5 | prssi 4353 | . . . . 5 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0) | |
6 | 3, 4, 5 | mp2an 708 | . . . 4 ⊢ {0, 1} ⊆ ℕ0 |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → {0, 1} ⊆ ℕ0) |
8 | 1, 2, 7 | brmptiunrelexpd 37975 | . 2 ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ ∃𝑛 ∈ {0, 1}𝐴(𝑅↑𝑟𝑛)𝐵)) |
9 | oveq2 6658 | . . . . 5 ⊢ (𝑛 = 0 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟0)) | |
10 | 9 | breqd 4664 | . . . 4 ⊢ (𝑛 = 0 → (𝐴(𝑅↑𝑟𝑛)𝐵 ↔ 𝐴(𝑅↑𝑟0)𝐵)) |
11 | oveq2 6658 | . . . . 5 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1)) | |
12 | 11 | breqd 4664 | . . . 4 ⊢ (𝑛 = 1 → (𝐴(𝑅↑𝑟𝑛)𝐵 ↔ 𝐴(𝑅↑𝑟1)𝐵)) |
13 | 10, 12 | rexprg 4235 | . . 3 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → (∃𝑛 ∈ {0, 1}𝐴(𝑅↑𝑟𝑛)𝐵 ↔ (𝐴(𝑅↑𝑟0)𝐵 ∨ 𝐴(𝑅↑𝑟1)𝐵))) |
14 | 3, 4, 13 | mp2an 708 | . 2 ⊢ (∃𝑛 ∈ {0, 1}𝐴(𝑅↑𝑟𝑛)𝐵 ↔ (𝐴(𝑅↑𝑟0)𝐵 ∨ 𝐴(𝑅↑𝑟1)𝐵)) |
15 | 8, 14 | syl6bb 276 | 1 ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ (𝐴(𝑅↑𝑟0)𝐵 ∨ 𝐴(𝑅↑𝑟1)𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 Vcvv 3200 ⊆ wss 3574 {cpr 4179 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 ℕ0cn0 11292 ↑𝑟crelexp 13760 r*crcl 37964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-seq 12802 df-relexp 13761 df-rcl 37965 |
This theorem is referenced by: brfvrcld2 37984 |
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