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Mirrors > Home > MPE Home > Th. List > rtrclreclem2 | Structured version Visualization version GIF version |
Description: The reflexive, transitive closure is indeed reflexive. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 13-Jul-2024.) |
Ref | Expression |
---|---|
rtrclreclem2.1 | ⊢ (𝜑 → Rel 𝑅) |
rtrclreclem2.2 | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
Ref | Expression |
---|---|
rtrclreclem2 | ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 12088 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
2 | ssid 3913 | . . . . . 6 ⊢ ( I ↾ ∪ ∪ 𝑅) ⊆ ( I ↾ ∪ ∪ 𝑅) | |
3 | rtrclreclem2.1 | . . . . . . 7 ⊢ (𝜑 → Rel 𝑅) | |
4 | rtrclreclem2.2 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
5 | 3, 4 | relexp0d 14570 | . . . . . 6 ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) |
6 | 2, 5 | sseqtrrid 3944 | . . . . 5 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟0)) |
7 | oveq2 7210 | . . . . . . 7 ⊢ (𝑛 = 0 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟0)) | |
8 | 7 | sseq2d 3923 | . . . . . 6 ⊢ (𝑛 = 0 → (( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟𝑛) ↔ ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟0))) |
9 | 8 | rspcev 3530 | . . . . 5 ⊢ ((0 ∈ ℕ0 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟0)) → ∃𝑛 ∈ ℕ0 ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟𝑛)) |
10 | 1, 6, 9 | sylancr 590 | . . . 4 ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟𝑛)) |
11 | ssiun 4945 | . . . 4 ⊢ (∃𝑛 ∈ ℕ0 ( I ↾ ∪ ∪ 𝑅) ⊆ (𝑅↑𝑟𝑛) → ( I ↾ ∪ ∪ 𝑅) ⊆ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)) |
13 | 4 | elexd 3421 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ V) |
14 | nn0ex 12079 | . . . . 5 ⊢ ℕ0 ∈ V | |
15 | ovex 7235 | . . . . 5 ⊢ (𝑅↑𝑟𝑛) ∈ V | |
16 | 14, 15 | iunex 7730 | . . . 4 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V |
17 | oveq1 7209 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛)) | |
18 | 17 | iuneq2d 4923 | . . . . 5 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)) |
19 | eqid 2734 | . . . . 5 ⊢ (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) | |
20 | 18, 19 | fvmptg 6805 | . . . 4 ⊢ ((𝑅 ∈ V ∧ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)) |
21 | 13, 16, 20 | sylancl 589 | . . 3 ⊢ (𝜑 → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)) |
22 | 12, 21 | sseqtrrd 3932 | . 2 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)) |
23 | df-rtrclrec 14602 | . . 3 ⊢ t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) | |
24 | fveq1 6705 | . . . . 5 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)) | |
25 | 24 | sseq2d 3923 | . . . 4 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → (( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅) ↔ ( I ↾ ∪ ∪ 𝑅) ⊆ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅))) |
26 | 25 | imbi2d 344 | . . 3 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → ((𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅)) ↔ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)))) |
27 | 23, 26 | ax-mp 5 | . 2 ⊢ ((𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅)) ↔ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅))) |
28 | 22, 27 | mpbir 234 | 1 ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2110 ∃wrex 3055 Vcvv 3401 ⊆ wss 3857 ∪ cuni 4809 ∪ ciun 4894 ↦ cmpt 5124 I cid 5443 ↾ cres 5542 Rel wrel 5545 ‘cfv 6369 (class class class)co 7202 0cc0 10712 ℕ0cn0 12073 ↑𝑟crelexp 14565 t*reccrtrcl 14601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-mulcl 10774 ax-i2m1 10780 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-nn 11814 df-n0 12074 df-relexp 14566 df-rtrclrec 14602 |
This theorem is referenced by: dfrtrcl2 14608 |
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