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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 42186 | . . 3 ⊢ r* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ {0, 1} (𝑟↑𝑟𝑛)) | |
2 | brfvrcld.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) | |
3 | 0nn0 12468 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
4 | 1nn0 12469 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
5 | prssi 4816 | . . . . 5 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0) | |
6 | 3, 4, 5 | mp2an 690 | . . . 4 ⊢ {0, 1} ⊆ ℕ0 |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → {0, 1} ⊆ ℕ0) |
8 | 1, 2, 7 | brmptiunrelexpd 42193 | . 2 ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ ∃𝑛 ∈ {0, 1}𝐴(𝑅↑𝑟𝑛)𝐵)) |
9 | oveq2 7400 | . . . . 5 ⊢ (𝑛 = 0 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟0)) | |
10 | 9 | breqd 5151 | . . . 4 ⊢ (𝑛 = 0 → (𝐴(𝑅↑𝑟𝑛)𝐵 ↔ 𝐴(𝑅↑𝑟0)𝐵)) |
11 | oveq2 7400 | . . . . 5 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1)) | |
12 | 11 | breqd 5151 | . . . 4 ⊢ (𝑛 = 1 → (𝐴(𝑅↑𝑟𝑛)𝐵 ↔ 𝐴(𝑅↑𝑟1)𝐵)) |
13 | 10, 12 | rexprg 4692 | . . 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 690 | . 2 ⊢ (∃𝑛 ∈ {0, 1}𝐴(𝑅↑𝑟𝑛)𝐵 ↔ (𝐴(𝑅↑𝑟0)𝐵 ∨ 𝐴(𝑅↑𝑟1)𝐵)) |
15 | 8, 14 | bitrdi 286 | 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 205 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ∃wrex 3069 Vcvv 3472 ⊆ wss 3943 {cpr 4623 class class class wbr 5140 ‘cfv 6531 (class class class)co 7392 0cc0 11091 1c1 11092 ℕ0cn0 12453 ↑𝑟crelexp 14947 r*crcl 42182 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5277 ax-sep 5291 ax-nul 5298 ax-pow 5355 ax-pr 5419 ax-un 7707 ax-cnex 11147 ax-resscn 11148 ax-1cn 11149 ax-icn 11150 ax-addcl 11151 ax-addrcl 11152 ax-mulcl 11153 ax-mulrcl 11154 ax-mulcom 11155 ax-addass 11156 ax-mulass 11157 ax-distr 11158 ax-i2m1 11159 ax-1ne0 11160 ax-1rid 11161 ax-rnegex 11162 ax-rrecex 11163 ax-cnre 11164 ax-pre-lttri 11165 ax-pre-lttrn 11166 ax-pre-ltadd 11167 ax-pre-mulgt0 11168 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3474 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4943 df-iun 4991 df-br 5141 df-opab 5203 df-mpt 5224 df-tr 5258 df-id 5566 df-eprel 5572 df-po 5580 df-so 5581 df-fr 5623 df-we 5625 df-xp 5674 df-rel 5675 df-cnv 5676 df-co 5677 df-dm 5678 df-rn 5679 df-res 5680 df-ima 5681 df-pred 6288 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7348 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7838 df-2nd 7957 df-frecs 8247 df-wrecs 8278 df-recs 8352 df-rdg 8391 df-er 8685 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11231 df-mnf 11232 df-xr 11233 df-ltxr 11234 df-le 11235 df-sub 11427 df-neg 11428 df-nn 12194 df-n0 12454 df-z 12540 df-uz 12804 df-seq 13948 df-relexp 14948 df-rcl 42183 |
This theorem is referenced by: brfvrcld2 42202 |
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