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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 40013 | . 2 ⊢ r* = (𝑟 ∈ V ↦ ((𝑟↑𝑟0) ∪ (𝑟↑𝑟1))) | |
2 | df-pr 4564 | . . . . 5 ⊢ {0, 1} = ({0} ∪ {1}) | |
3 | iuneq1 4928 | . . . . 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 5009 | . . . 4 ⊢ ∪ 𝑛 ∈ ({0} ∪ {1})(𝑟↑𝑟𝑛) = (∪ 𝑛 ∈ {0} (𝑟↑𝑟𝑛) ∪ ∪ 𝑛 ∈ {1} (𝑟↑𝑟𝑛)) | |
6 | c0ex 10629 | . . . . . 6 ⊢ 0 ∈ V | |
7 | oveq2 7158 | . . . . . 6 ⊢ (𝑛 = 0 → (𝑟↑𝑟𝑛) = (𝑟↑𝑟0)) | |
8 | 6, 7 | iunxsn 5006 | . . . . 5 ⊢ ∪ 𝑛 ∈ {0} (𝑟↑𝑟𝑛) = (𝑟↑𝑟0) |
9 | 1ex 10631 | . . . . . 6 ⊢ 1 ∈ V | |
10 | oveq2 7158 | . . . . . 6 ⊢ (𝑛 = 1 → (𝑟↑𝑟𝑛) = (𝑟↑𝑟1)) | |
11 | 9, 10 | iunxsn 5006 | . . . . 5 ⊢ ∪ 𝑛 ∈ {1} (𝑟↑𝑟𝑛) = (𝑟↑𝑟1) |
12 | 8, 11 | uneq12i 4137 | . . . 4 ⊢ (∪ 𝑛 ∈ {0} (𝑟↑𝑟𝑛) ∪ ∪ 𝑛 ∈ {1} (𝑟↑𝑟𝑛)) = ((𝑟↑𝑟0) ∪ (𝑟↑𝑟1)) |
13 | 4, 5, 12 | 3eqtri 2848 | . . 3 ⊢ ∪ 𝑛 ∈ {0, 1} (𝑟↑𝑟𝑛) = ((𝑟↑𝑟0) ∪ (𝑟↑𝑟1)) |
14 | 13 | mpteq2i 5151 | . 2 ⊢ (𝑟 ∈ V ↦ ∪ 𝑛 ∈ {0, 1} (𝑟↑𝑟𝑛)) = (𝑟 ∈ V ↦ ((𝑟↑𝑟0) ∪ (𝑟↑𝑟1))) |
15 | 1, 14 | eqtr4i 2847 | 1 ⊢ r* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ {0, 1} (𝑟↑𝑟𝑛)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 Vcvv 3495 ∪ cun 3934 {csn 4561 {cpr 4563 ∪ ciun 4912 ↦ cmpt 5139 (class class class)co 7150 0cc0 10531 1c1 10532 ↑𝑟crelexp 14373 r*crcl 40010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-seq 13364 df-relexp 14374 df-rcl 40011 |
This theorem is referenced by: brfvrcld 40029 fvrcllb0d 40031 fvrcllb0da 40032 fvrcllb1d 40033 corclrcl 40045 corcltrcl 40077 cotrclrcl 40080 |
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