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Mirrors > Home > MPE Home > Th. List > dfrtrclrec2 | Structured version Visualization version GIF version |
Description: If two elements are connected by a reflexive, transitive closure, then they are connected via 𝑛 instances the relation, for some 𝑛. (Contributed by Drahflow, 12-Nov-2015.) |
Ref | Expression |
---|---|
rtrclreclem.1 | ⊢ (𝜑 → Rel 𝑅) |
rtrclreclem.2 | ⊢ (𝜑 → 𝑅 ∈ V) |
Ref | Expression |
---|---|
dfrtrclrec2 | ⊢ (𝜑 → (𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rtrclreclem.2 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ V) | |
2 | nn0ex 11897 | . . . . 5 ⊢ ℕ0 ∈ V | |
3 | ovex 7183 | . . . . 5 ⊢ (𝑅↑𝑟𝑛) ∈ V | |
4 | 2, 3 | iunex 7663 | . . . 4 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V |
5 | oveq1 7157 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛)) | |
6 | 5 | iuneq2d 4941 | . . . . 5 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)) |
7 | eqid 2821 | . . . . 5 ⊢ (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) | |
8 | 6, 7 | fvmptg 6761 | . . . 4 ⊢ ((𝑅 ∈ V ∧ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)) |
9 | 1, 4, 8 | sylancl 588 | . . 3 ⊢ (𝜑 → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)) |
10 | breq 5061 | . . . 4 ⊢ (((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) → (𝐴((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)𝐵 ↔ 𝐴∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)𝐵)) | |
11 | eliun 4916 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 ∈ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ↔ ∃𝑛 ∈ ℕ0 〈𝐴, 𝐵〉 ∈ (𝑅↑𝑟𝑛)) | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ↔ ∃𝑛 ∈ ℕ0 〈𝐴, 𝐵〉 ∈ (𝑅↑𝑟𝑛))) |
13 | df-br 5060 | . . . . 5 ⊢ (𝐴∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)) | |
14 | df-br 5060 | . . . . . 6 ⊢ (𝐴(𝑅↑𝑟𝑛)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅↑𝑟𝑛)) | |
15 | 14 | rexbii 3247 | . . . . 5 ⊢ (∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵 ↔ ∃𝑛 ∈ ℕ0 〈𝐴, 𝐵〉 ∈ (𝑅↑𝑟𝑛)) |
16 | 12, 13, 15 | 3bitr4g 316 | . . . 4 ⊢ (𝜑 → (𝐴∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵)) |
17 | 10, 16 | sylan9bb 512 | . . 3 ⊢ ((((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∧ 𝜑) → (𝐴((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵)) |
18 | 9, 17 | mpancom 686 | . 2 ⊢ (𝜑 → (𝐴((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵)) |
19 | df-rtrclrec 14409 | . . 3 ⊢ t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) | |
20 | fveq1 6664 | . . . . . 6 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)) | |
21 | 20 | breqd 5070 | . . . . 5 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → (𝐴(t*rec‘𝑅)𝐵 ↔ 𝐴((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)𝐵)) |
22 | 21 | bibi1d 346 | . . . 4 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → ((𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵) ↔ (𝐴((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵))) |
23 | 22 | imbi2d 343 | . . 3 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → ((𝜑 → (𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵)) ↔ (𝜑 → (𝐴((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵)))) |
24 | 19, 23 | ax-mp 5 | . 2 ⊢ ((𝜑 → (𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵)) ↔ (𝜑 → (𝐴((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵))) |
25 | 18, 24 | mpbir 233 | 1 ⊢ (𝜑 → (𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 Vcvv 3495 〈cop 4567 ∪ ciun 4912 class class class wbr 5059 ↦ cmpt 5139 Rel wrel 5555 ‘cfv 6350 (class class class)co 7150 ℕ0cn0 11891 ↑𝑟crelexp 14373 t*reccrtrcl 14408 |
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-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-1cn 10589 ax-addcl 10591 |
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-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-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-ov 7153 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-nn 11633 df-n0 11892 df-rtrclrec 14409 |
This theorem is referenced by: rtrclreclem3 14413 rtrclind 14418 |
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