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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 3459 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
2 | nnnn0 11892 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
3 | relexpcnv 14386 | . . . . . . 7 ⊢ ((𝑛 ∈ ℕ0 ∧ 𝑅 ∈ V) → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) | |
4 | 2, 3 | sylan 583 | . . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑅 ∈ V) → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) |
5 | 4 | expcom 417 | . . . . 5 ⊢ (𝑅 ∈ V → (𝑛 ∈ ℕ → ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛))) |
6 | 5 | ralrimiv 3148 | . . . 4 ⊢ (𝑅 ∈ V → ∀𝑛 ∈ ℕ ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) |
7 | iuneq2 4900 | . . . 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 7142 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛)) | |
10 | 9 | iuneq2d 4910 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛)) |
11 | dftrcl3 40421 | . . . . . 6 ⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛)) | |
12 | nnex 11631 | . . . . . . 7 ⊢ ℕ ∈ V | |
13 | ovex 7168 | . . . . . . 7 ⊢ (𝑅↑𝑟𝑛) ∈ V | |
14 | 12, 13 | iunex 7651 | . . . . . 6 ⊢ ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ∈ V |
15 | 10, 11, 14 | fvmpt 6745 | . . . . 5 ⊢ (𝑅 ∈ V → (t+‘𝑅) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛)) |
16 | 15 | cnveqd 5710 | . . . 4 ⊢ (𝑅 ∈ V → ◡(t+‘𝑅) = ◡∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛)) |
17 | cnviun 40351 | . . . 4 ⊢ ◡∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ ◡(𝑅↑𝑟𝑛) | |
18 | 16, 17 | eqtrdi 2849 | . . 3 ⊢ (𝑅 ∈ V → ◡(t+‘𝑅) = ∪ 𝑛 ∈ ℕ ◡(𝑅↑𝑟𝑛)) |
19 | cnvexg 7611 | . . . 4 ⊢ (𝑅 ∈ V → ◡𝑅 ∈ V) | |
20 | oveq1 7142 | . . . . . 6 ⊢ (𝑠 = ◡𝑅 → (𝑠↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) | |
21 | 20 | iuneq2d 4910 | . . . . 5 ⊢ (𝑠 = ◡𝑅 → ∪ 𝑛 ∈ ℕ (𝑠↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ (◡𝑅↑𝑟𝑛)) |
22 | dftrcl3 40421 | . . . . 5 ⊢ t+ = (𝑠 ∈ V ↦ ∪ 𝑛 ∈ ℕ (𝑠↑𝑟𝑛)) | |
23 | ovex 7168 | . . . . . 6 ⊢ (◡𝑅↑𝑟𝑛) ∈ V | |
24 | 12, 23 | iunex 7651 | . . . . 5 ⊢ ∪ 𝑛 ∈ ℕ (◡𝑅↑𝑟𝑛) ∈ V |
25 | 21, 22, 24 | fvmpt 6745 | . . . 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 2843 | . 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 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ∪ ciun 4881 ◡ccnv 5518 ‘cfv 6324 (class class class)co 7135 ℕcn 11625 ℕ0cn0 11885 t+ctcl 14336 ↑𝑟crelexp 14370 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-seq 13365 df-trcl 14338 df-relexp 14371 |
This theorem is referenced by: rntrclfvRP 40432 |
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