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Mathbox for Richard Penner |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvtrclfv | Structured version Visualization version GIF version |
Description: The converse of the transitive closure is equal to the transitive closure of the converse relation. (Contributed by RP, 19-Jul-2020.) |
Ref | Expression |
---|---|
cnvtrclfv | ⊢ (𝑅 ∈ 𝑉 → ◡(t+‘𝑅) = (t+‘◡𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3492 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
2 | nnnn0 12475 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
3 | relexpcnv 14978 | . . . . . . 7 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑅 ∈ V) → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) | |
4 | 2, 3 | sylan 580 | . . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑅 ∈ V) → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) |
5 | 4 | expcom 414 | . . . . 5 ⊢ (𝑅 ∈ V → (𝑛 ∈ ℕ → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛))) |
6 | 5 | ralrimiv 3145 | . . . 4 ⊢ (𝑅 ∈ V → ∀𝑛 ∈ ℕ ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) |
7 | iuneq2 5015 | . . . 4 ⊢ (∀𝑛 ∈ ℕ ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛) → ∪ 𝑛 ∈ ℕ ◡(𝑅↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ (◡𝑅↑𝑟𝑛)) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝑅 ∈ V → ∪ 𝑛 ∈ ℕ ◡(𝑅↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ (◡𝑅↑𝑟𝑛)) |
9 | oveq1 7412 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛)) | |
10 | 9 | iuneq2d 5025 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛)) |
11 | dftrcl3 42456 | . . . . . 6 ⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛)) | |
12 | nnex 12214 | . . . . . . 7 ⊢ ℕ ∈ V | |
13 | ovex 7438 | . . . . . . 7 ⊢ (𝑅↑𝑟𝑛) ∈ V | |
14 | 12, 13 | iunex 7951 | . . . . . 6 ⊢ ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ∈ V |
15 | 10, 11, 14 | fvmpt 6995 | . . . . 5 ⊢ (𝑅 ∈ V → (t+‘𝑅) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛)) |
16 | 15 | cnveqd 5873 | . . . 4 ⊢ (𝑅 ∈ V → ◡(t+‘𝑅) = ◡∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛)) |
17 | cnviun 42386 | . . . 4 ⊢ ◡∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ ◡(𝑅↑𝑟𝑛) | |
18 | 16, 17 | eqtrdi 2788 | . . 3 ⊢ (𝑅 ∈ V → ◡(t+‘𝑅) = ∪ 𝑛 ∈ ℕ ◡(𝑅↑𝑟𝑛)) |
19 | cnvexg 7911 | . . . 4 ⊢ (𝑅 ∈ V → ◡𝑅 ∈ V) | |
20 | oveq1 7412 | . . . . . 6 ⊢ (𝑠 = ◡𝑅 → (𝑠↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) | |
21 | 20 | iuneq2d 5025 | . . . . 5 ⊢ (𝑠 = ◡𝑅 → ∪ 𝑛 ∈ ℕ (𝑠↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ (◡𝑅↑𝑟𝑛)) |
22 | dftrcl3 42456 | . . . . 5 ⊢ t+ = (𝑠 ∈ V ↦ ∪ 𝑛 ∈ ℕ (𝑠↑𝑟𝑛)) | |
23 | ovex 7438 | . . . . . 6 ⊢ (◡𝑅↑𝑟𝑛) ∈ V | |
24 | 12, 23 | iunex 7951 | . . . . 5 ⊢ ∪ 𝑛 ∈ ℕ (◡𝑅↑𝑟𝑛) ∈ V |
25 | 21, 22, 24 | fvmpt 6995 | . . . 4 ⊢ (◡𝑅 ∈ V → (t+‘◡𝑅) = ∪ 𝑛 ∈ ℕ (◡𝑅↑𝑟𝑛)) |
26 | 19, 25 | syl 17 | . . 3 ⊢ (𝑅 ∈ V → (t+‘◡𝑅) = ∪ 𝑛 ∈ ℕ (◡𝑅↑𝑟𝑛)) |
27 | 8, 18, 26 | 3eqtr4d 2782 | . 2 ⊢ (𝑅 ∈ V → ◡(t+‘𝑅) = (t+‘◡𝑅)) |
28 | 1, 27 | syl 17 | 1 ⊢ (𝑅 ∈ 𝑉 → ◡(t+‘𝑅) = (t+‘◡𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ∪ ciun 4996 ◡ccnv 5674 ‘cfv 6540 (class class class)co 7405 ℕcn 12208 ℕ0cn0 12468 t+ctcl 14928 ↑𝑟crelexp 14962 |
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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-seq 13963 df-trcl 14930 df-relexp 14963 |
This theorem is referenced by: rntrclfvRP 42467 |
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