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Mirrors > Home > MPE Home > Th. List > relexp1g | Structured version Visualization version GIF version |
Description: A relation composed once is itself. (Contributed by RP, 22-May-2020.) |
Ref | Expression |
---|---|
relexp1g | ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-relexp 13761 | . . 3 ⊢ ↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑅 ∈ 𝑉 → ↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))) |
3 | simprr 796 | . . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑟 = 𝑅 ∧ 𝑛 = 1)) → 𝑛 = 1) | |
4 | ax-1ne0 10005 | . . . . . . 7 ⊢ 1 ≠ 0 | |
5 | neeq1 2856 | . . . . . . 7 ⊢ (𝑛 = 1 → (𝑛 ≠ 0 ↔ 1 ≠ 0)) | |
6 | 4, 5 | mpbiri 248 | . . . . . 6 ⊢ (𝑛 = 1 → 𝑛 ≠ 0) |
7 | 3, 6 | syl 17 | . . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑟 = 𝑅 ∧ 𝑛 = 1)) → 𝑛 ≠ 0) |
8 | 7 | neneqd 2799 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑟 = 𝑅 ∧ 𝑛 = 1)) → ¬ 𝑛 = 0) |
9 | 8 | iffalsed 4097 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑟 = 𝑅 ∧ 𝑛 = 1)) → if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)) = (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)) |
10 | simprl 794 | . . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑟 = 𝑅 ∧ 𝑛 = 1)) → 𝑟 = 𝑅) | |
11 | 10 | mpteq2dv 4745 | . . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑟 = 𝑅 ∧ 𝑛 = 1)) → (𝑧 ∈ V ↦ 𝑟) = (𝑧 ∈ V ↦ 𝑅)) |
12 | 11 | seqeq3d 12809 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑟 = 𝑅 ∧ 𝑛 = 1)) → seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟)) = seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑅))) |
13 | 12, 3 | fveq12d 6197 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑟 = 𝑅 ∧ 𝑛 = 1)) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛) = (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑅))‘1)) |
14 | 1z 11407 | . . . 4 ⊢ 1 ∈ ℤ | |
15 | eqidd 2623 | . . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑟 = 𝑅 ∧ 𝑛 = 1)) → (𝑧 ∈ V ↦ 𝑅) = (𝑧 ∈ V ↦ 𝑅)) | |
16 | eqidd 2623 | . . . . 5 ⊢ (((𝑅 ∈ 𝑉 ∧ (𝑟 = 𝑅 ∧ 𝑛 = 1)) ∧ 𝑧 = 1) → 𝑅 = 𝑅) | |
17 | 1ex 10035 | . . . . . 6 ⊢ 1 ∈ V | |
18 | 17 | a1i 11 | . . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑟 = 𝑅 ∧ 𝑛 = 1)) → 1 ∈ V) |
19 | simpl 473 | . . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑟 = 𝑅 ∧ 𝑛 = 1)) → 𝑅 ∈ 𝑉) | |
20 | 15, 16, 18, 19 | fvmptd 6288 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑟 = 𝑅 ∧ 𝑛 = 1)) → ((𝑧 ∈ V ↦ 𝑅)‘1) = 𝑅) |
21 | 14, 20 | seq1i 12815 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑟 = 𝑅 ∧ 𝑛 = 1)) → (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑅))‘1) = 𝑅) |
22 | 9, 13, 21 | 3eqtrd 2660 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑟 = 𝑅 ∧ 𝑛 = 1)) → if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)) = 𝑅) |
23 | elex 3212 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
24 | 1nn0 11308 | . . 3 ⊢ 1 ∈ ℕ0 | |
25 | 24 | a1i 11 | . 2 ⊢ (𝑅 ∈ 𝑉 → 1 ∈ ℕ0) |
26 | 2, 22, 23, 25, 23 | ovmpt2d 6788 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∪ cun 3572 ifcif 4086 ↦ cmpt 4729 I cid 5023 dom cdm 5114 ran crn 5115 ↾ cres 5116 ∘ ccom 5118 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 0cc0 9936 1c1 9937 ℕ0cn0 11292 seqcseq 12801 ↑𝑟crelexp 13760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-seq 12802 df-relexp 13761 |
This theorem is referenced by: dfid5 13767 dfid6 13768 relexpsucr 13769 relexp1d 13771 relexpsucnnl 13772 relexpsucl 13773 relexpcnv 13775 relexprelg 13778 relexpnndm 13781 relexpfld 13789 relexpaddnn 13791 relexpaddg 13793 dfrcl3 37967 relexp2 37969 iunrelexp0 37994 relexpxpnnidm 37995 corclrcl 37999 iunrelexpmin1 38000 trclrelexplem 38003 iunrelexpmin2 38004 relexp01min 38005 relexp0a 38008 relexpaddss 38010 dftrcl3 38012 cotrcltrcl 38017 trclimalb2 38018 trclfvdecomr 38020 dfrtrcl3 38025 corcltrcl 38031 cotrclrcl 38034 |
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