Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > dfrtrcl4 | Structured version Visualization version GIF version |
Description: Reflexive-transitive closure of a relation, expressed as the union of the zeroth power and the transitive closure. (Contributed by RP, 5-Jun-2020.) |
Ref | Expression |
---|---|
dfrtrcl4 | ⊢ t* = (𝑟 ∈ V ↦ ((𝑟↑𝑟0) ∪ (t+‘𝑟))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrtrcl3 39956 | . 2 ⊢ t* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) | |
2 | df-n0 11886 | . . . . . . 7 ⊢ ℕ0 = (ℕ ∪ {0}) | |
3 | 2 | equncomi 4128 | . . . . . 6 ⊢ ℕ0 = ({0} ∪ ℕ) |
4 | iuneq1 4926 | . . . . . 6 ⊢ (ℕ0 = ({0} ∪ ℕ) → ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ({0} ∪ ℕ)(𝑟↑𝑟𝑛)) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ({0} ∪ ℕ)(𝑟↑𝑟𝑛) |
6 | iunxun 5007 | . . . . 5 ⊢ ∪ 𝑛 ∈ ({0} ∪ ℕ)(𝑟↑𝑟𝑛) = (∪ 𝑛 ∈ {0} (𝑟↑𝑟𝑛) ∪ ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛)) | |
7 | 5, 6 | eqtri 2841 | . . . 4 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) = (∪ 𝑛 ∈ {0} (𝑟↑𝑟𝑛) ∪ ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛)) |
8 | c0ex 10623 | . . . . . . 7 ⊢ 0 ∈ V | |
9 | oveq2 7153 | . . . . . . 7 ⊢ (𝑛 = 0 → (𝑟↑𝑟𝑛) = (𝑟↑𝑟0)) | |
10 | 8, 9 | iunxsn 5004 | . . . . . 6 ⊢ ∪ 𝑛 ∈ {0} (𝑟↑𝑟𝑛) = (𝑟↑𝑟0) |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝑟 ∈ V → ∪ 𝑛 ∈ {0} (𝑟↑𝑟𝑛) = (𝑟↑𝑟0)) |
12 | oveq1 7152 | . . . . . . . 8 ⊢ (𝑥 = 𝑟 → (𝑥↑𝑟𝑛) = (𝑟↑𝑟𝑛)) | |
13 | 12 | iuneq2d 4939 | . . . . . . 7 ⊢ (𝑥 = 𝑟 → ∪ 𝑛 ∈ ℕ (𝑥↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛)) |
14 | dftrcl3 39943 | . . . . . . 7 ⊢ t+ = (𝑥 ∈ V ↦ ∪ 𝑛 ∈ ℕ (𝑥↑𝑟𝑛)) | |
15 | nnex 11632 | . . . . . . . 8 ⊢ ℕ ∈ V | |
16 | ovex 7178 | . . . . . . . 8 ⊢ (𝑟↑𝑟𝑛) ∈ V | |
17 | 15, 16 | iunex 7658 | . . . . . . 7 ⊢ ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛) ∈ V |
18 | 13, 14, 17 | fvmpt 6761 | . . . . . 6 ⊢ (𝑟 ∈ V → (t+‘𝑟) = ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛)) |
19 | 18 | eqcomd 2824 | . . . . 5 ⊢ (𝑟 ∈ V → ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛) = (t+‘𝑟)) |
20 | 11, 19 | uneq12d 4137 | . . . 4 ⊢ (𝑟 ∈ V → (∪ 𝑛 ∈ {0} (𝑟↑𝑟𝑛) ∪ ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛)) = ((𝑟↑𝑟0) ∪ (t+‘𝑟))) |
21 | 7, 20 | syl5eq 2865 | . . 3 ⊢ (𝑟 ∈ V → ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) = ((𝑟↑𝑟0) ∪ (t+‘𝑟))) |
22 | 21 | mpteq2ia 5148 | . 2 ⊢ (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) = (𝑟 ∈ V ↦ ((𝑟↑𝑟0) ∪ (t+‘𝑟))) |
23 | 1, 22 | eqtri 2841 | 1 ⊢ t* = (𝑟 ∈ V ↦ ((𝑟↑𝑟0) ∪ (t+‘𝑟))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∪ cun 3931 {csn 4557 ∪ ciun 4910 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 0cc0 10525 ℕcn 11626 ℕ0cn0 11885 t+ctcl 14333 t*crtcl 14334 ↑𝑟crelexp 14367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-seq 13358 df-trcl 14335 df-rtrcl 14336 df-relexp 14368 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |