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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfrcl4 | Structured version Visualization version GIF version | ||
| Description: Reflexive closure of a relation as indexed union of powers of the relation. (Contributed by RP, 8-Jun-2020.) |
| Ref | Expression |
|---|---|
| dfrcl4 | ⊢ r* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ {0, 1} (𝑟↑𝑟𝑛)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfrcl3 43688 | . 2 ⊢ r* = (𝑟 ∈ V ↦ ((𝑟↑𝑟0) ∪ (𝑟↑𝑟1))) | |
| 2 | df-pr 4629 | . . . . 5 ⊢ {0, 1} = ({0} ∪ {1}) | |
| 3 | iuneq1 5008 | . . . . 5 ⊢ ({0, 1} = ({0} ∪ {1}) → ∪ 𝑛 ∈ {0, 1} (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ({0} ∪ {1})(𝑟↑𝑟𝑛)) | |
| 4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ∪ 𝑛 ∈ {0, 1} (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ({0} ∪ {1})(𝑟↑𝑟𝑛) |
| 5 | iunxun 5094 | . . . 4 ⊢ ∪ 𝑛 ∈ ({0} ∪ {1})(𝑟↑𝑟𝑛) = (∪ 𝑛 ∈ {0} (𝑟↑𝑟𝑛) ∪ ∪ 𝑛 ∈ {1} (𝑟↑𝑟𝑛)) | |
| 6 | c0ex 11255 | . . . . . 6 ⊢ 0 ∈ V | |
| 7 | oveq2 7439 | . . . . . 6 ⊢ (𝑛 = 0 → (𝑟↑𝑟𝑛) = (𝑟↑𝑟0)) | |
| 8 | 6, 7 | iunxsn 5091 | . . . . 5 ⊢ ∪ 𝑛 ∈ {0} (𝑟↑𝑟𝑛) = (𝑟↑𝑟0) |
| 9 | 1ex 11257 | . . . . . 6 ⊢ 1 ∈ V | |
| 10 | oveq2 7439 | . . . . . 6 ⊢ (𝑛 = 1 → (𝑟↑𝑟𝑛) = (𝑟↑𝑟1)) | |
| 11 | 9, 10 | iunxsn 5091 | . . . . 5 ⊢ ∪ 𝑛 ∈ {1} (𝑟↑𝑟𝑛) = (𝑟↑𝑟1) |
| 12 | 8, 11 | uneq12i 4166 | . . . 4 ⊢ (∪ 𝑛 ∈ {0} (𝑟↑𝑟𝑛) ∪ ∪ 𝑛 ∈ {1} (𝑟↑𝑟𝑛)) = ((𝑟↑𝑟0) ∪ (𝑟↑𝑟1)) |
| 13 | 4, 5, 12 | 3eqtri 2769 | . . 3 ⊢ ∪ 𝑛 ∈ {0, 1} (𝑟↑𝑟𝑛) = ((𝑟↑𝑟0) ∪ (𝑟↑𝑟1)) |
| 14 | 13 | mpteq2i 5247 | . 2 ⊢ (𝑟 ∈ V ↦ ∪ 𝑛 ∈ {0, 1} (𝑟↑𝑟𝑛)) = (𝑟 ∈ V ↦ ((𝑟↑𝑟0) ∪ (𝑟↑𝑟1))) |
| 15 | 1, 14 | eqtr4i 2768 | 1 ⊢ r* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ {0, 1} (𝑟↑𝑟𝑛)) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 Vcvv 3480 ∪ cun 3949 {csn 4626 {cpr 4628 ∪ ciun 4991 ↦ cmpt 5225 (class class class)co 7431 0cc0 11155 1c1 11156 ↑𝑟crelexp 15058 r*crcl 43685 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-seq 14043 df-relexp 15059 df-rcl 43686 |
| This theorem is referenced by: brfvrcld 43704 fvrcllb0d 43706 fvrcllb0da 43707 fvrcllb1d 43708 corclrcl 43720 corcltrcl 43752 cotrclrcl 43755 |
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