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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 3416 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
2 | nnnn0 11984 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
3 | relexpcnv 14485 | . . . . . . 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 3095 | . . . 4 ⊢ (𝑅 ∈ V → ∀𝑛 ∈ ℕ ◡(𝑅↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) |
7 | iuneq2 4901 | . . . 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 7178 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛)) | |
10 | 9 | iuneq2d 4911 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛)) |
11 | dftrcl3 40866 | . . . . . 6 ⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛)) | |
12 | nnex 11723 | . . . . . . 7 ⊢ ℕ ∈ V | |
13 | ovex 7204 | . . . . . . 7 ⊢ (𝑅↑𝑟𝑛) ∈ V | |
14 | 12, 13 | iunex 7695 | . . . . . 6 ⊢ ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ∈ V |
15 | 10, 11, 14 | fvmpt 6776 | . . . . 5 ⊢ (𝑅 ∈ V → (t+‘𝑅) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛)) |
16 | 15 | cnveqd 5719 | . . . 4 ⊢ (𝑅 ∈ V → ◡(t+‘𝑅) = ◡∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛)) |
17 | cnviun 40796 | . . . 4 ⊢ ◡∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ ◡(𝑅↑𝑟𝑛) | |
18 | 16, 17 | eqtrdi 2789 | . . 3 ⊢ (𝑅 ∈ V → ◡(t+‘𝑅) = ∪ 𝑛 ∈ ℕ ◡(𝑅↑𝑟𝑛)) |
19 | cnvexg 7656 | . . . 4 ⊢ (𝑅 ∈ V → ◡𝑅 ∈ V) | |
20 | oveq1 7178 | . . . . . 6 ⊢ (𝑠 = ◡𝑅 → (𝑠↑𝑟𝑛) = (◡𝑅↑𝑟𝑛)) | |
21 | 20 | iuneq2d 4911 | . . . . 5 ⊢ (𝑠 = ◡𝑅 → ∪ 𝑛 ∈ ℕ (𝑠↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ (◡𝑅↑𝑟𝑛)) |
22 | dftrcl3 40866 | . . . . 5 ⊢ t+ = (𝑠 ∈ V ↦ ∪ 𝑛 ∈ ℕ (𝑠↑𝑟𝑛)) | |
23 | ovex 7204 | . . . . . 6 ⊢ (◡𝑅↑𝑟𝑛) ∈ V | |
24 | 12, 23 | iunex 7695 | . . . . 5 ⊢ ∪ 𝑛 ∈ ℕ (◡𝑅↑𝑟𝑛) ∈ V |
25 | 21, 22, 24 | fvmpt 6776 | . . . 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 2783 | . 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 1542 ∈ wcel 2113 ∀wral 3053 Vcvv 3398 ∪ ciun 4882 ◡ccnv 5525 ‘cfv 6340 (class class class)co 7171 ℕcn 11717 ℕ0cn0 11977 t+ctcl 14435 ↑𝑟crelexp 14469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 ax-cnex 10672 ax-resscn 10673 ax-1cn 10674 ax-icn 10675 ax-addcl 10676 ax-addrcl 10677 ax-mulcl 10678 ax-mulrcl 10679 ax-mulcom 10680 ax-addass 10681 ax-mulass 10682 ax-distr 10683 ax-i2m1 10684 ax-1ne0 10685 ax-1rid 10686 ax-rnegex 10687 ax-rrecex 10688 ax-cnre 10689 ax-pre-lttri 10690 ax-pre-lttrn 10691 ax-pre-ltadd 10692 ax-pre-mulgt0 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7128 df-ov 7174 df-oprab 7175 df-mpo 7176 df-om 7601 df-2nd 7716 df-wrecs 7977 df-recs 8038 df-rdg 8076 df-er 8321 df-en 8557 df-dom 8558 df-sdom 8559 df-pnf 10756 df-mnf 10757 df-xr 10758 df-ltxr 10759 df-le 10760 df-sub 10951 df-neg 10952 df-nn 11718 df-2 11780 df-n0 11978 df-z 12064 df-uz 12326 df-seq 13462 df-trcl 14437 df-relexp 14470 |
This theorem is referenced by: rntrclfvRP 40877 |
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