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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfrcl3 | Structured version Visualization version GIF version |
Description: Reflexive closure of a relation as union of powers of the relation. (Contributed by RP, 6-Jun-2020.) |
Ref | Expression |
---|---|
dfrcl3 | ⊢ r* = (𝑥 ∈ V ↦ ((𝑥↑𝑟0) ∪ (𝑥↑𝑟1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrcl2 38753 | . 2 ⊢ r* = (𝑥 ∈ V ↦ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥)) | |
2 | relexp0g 14107 | . . . 4 ⊢ (𝑥 ∈ V → (𝑥↑𝑟0) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))) | |
3 | relexp1g 14111 | . . . 4 ⊢ (𝑥 ∈ V → (𝑥↑𝑟1) = 𝑥) | |
4 | 2, 3 | uneq12d 3970 | . . 3 ⊢ (𝑥 ∈ V → ((𝑥↑𝑟0) ∪ (𝑥↑𝑟1)) = (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥)) |
5 | 4 | mpteq2ia 4937 | . 2 ⊢ (𝑥 ∈ V ↦ ((𝑥↑𝑟0) ∪ (𝑥↑𝑟1))) = (𝑥 ∈ V ↦ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥)) |
6 | 1, 5 | eqtr4i 2828 | 1 ⊢ r* = (𝑥 ∈ V ↦ ((𝑥↑𝑟0) ∪ (𝑥↑𝑟1))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 Vcvv 3389 ∪ cun 3771 ↦ cmpt 4926 I cid 5223 dom cdm 5316 ran crn 5317 ↾ cres 5318 (class class class)co 6882 0cc0 10228 1c1 10229 ↑𝑟crelexp 14105 r*crcl 38751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2379 ax-ext 2781 ax-sep 4979 ax-nul 4987 ax-pow 5039 ax-pr 5101 ax-un 7187 ax-cnex 10284 ax-resscn 10285 ax-1cn 10286 ax-icn 10287 ax-addcl 10288 ax-addrcl 10289 ax-mulcl 10290 ax-mulrcl 10291 ax-mulcom 10292 ax-addass 10293 ax-mulass 10294 ax-distr 10295 ax-i2m1 10296 ax-1ne0 10297 ax-1rid 10298 ax-rnegex 10299 ax-rrecex 10300 ax-cnre 10301 ax-pre-lttri 10302 ax-pre-lttrn 10303 ax-pre-ltadd 10304 ax-pre-mulgt0 10305 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2593 df-eu 2611 df-clab 2790 df-cleq 2796 df-clel 2799 df-nfc 2934 df-ne 2976 df-nel 3079 df-ral 3098 df-rex 3099 df-reu 3100 df-rab 3102 df-v 3391 df-sbc 3638 df-csb 3733 df-dif 3776 df-un 3778 df-in 3780 df-ss 3787 df-pss 3789 df-nul 4120 df-if 4282 df-pw 4355 df-sn 4373 df-pr 4375 df-tp 4377 df-op 4379 df-uni 4633 df-int 4672 df-iun 4716 df-br 4848 df-opab 4910 df-mpt 4927 df-tr 4950 df-id 5224 df-eprel 5229 df-po 5237 df-so 5238 df-fr 5275 df-we 5277 df-xp 5322 df-rel 5323 df-cnv 5324 df-co 5325 df-dm 5326 df-rn 5327 df-res 5328 df-ima 5329 df-pred 5902 df-ord 5948 df-on 5949 df-lim 5950 df-suc 5951 df-iota 6068 df-fun 6107 df-fn 6108 df-f 6109 df-f1 6110 df-fo 6111 df-f1o 6112 df-fv 6113 df-riota 6843 df-ov 6885 df-oprab 6886 df-mpt2 6887 df-om 7304 df-2nd 7406 df-wrecs 7649 df-recs 7711 df-rdg 7749 df-er 7986 df-en 8200 df-dom 8201 df-sdom 8202 df-pnf 10369 df-mnf 10370 df-xr 10371 df-ltxr 10372 df-le 10373 df-sub 10562 df-neg 10563 df-nn 11317 df-n0 11585 df-z 11671 df-uz 11935 df-seq 13060 df-relexp 14106 df-rcl 38752 |
This theorem is referenced by: dfrcl4 38755 |
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