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Mirrors > Home > MPE Home > Th. List > brtrclfv | Structured version Visualization version GIF version |
Description: Two ways of expressing the transitive closure of a binary relation. (Contributed by RP, 9-May-2020.) |
Ref | Expression |
---|---|
brtrclfv | ⊢ (𝑅 ∈ 𝑉 → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trclfv 14182 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}) | |
2 | 1 | breqd 4967 | . 2 ⊢ (𝑅 ∈ 𝑉 → (𝐴(t+‘𝑅)𝐵 ↔ 𝐴∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝐵)) |
3 | brintclab 14183 | . . 3 ⊢ (𝐴∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝐵 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 〈𝐴, 𝐵〉 ∈ 𝑟)) | |
4 | df-br 4957 | . . . . 5 ⊢ (𝐴𝑟𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑟) | |
5 | 4 | imbi2i 337 | . . . 4 ⊢ (((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵) ↔ ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 〈𝐴, 𝐵〉 ∈ 𝑟)) |
6 | 5 | albii 1799 | . . 3 ⊢ (∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵) ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 〈𝐴, 𝐵〉 ∈ 𝑟)) |
7 | 3, 6 | bitr4i 279 | . 2 ⊢ (𝐴∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}𝐵 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵)) |
8 | 2, 7 | syl6bb 288 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1518 ∈ wcel 2079 {cab 2773 ⊆ wss 3854 〈cop 4472 ∩ cint 4776 class class class wbr 4956 ∘ ccom 5439 ‘cfv 6217 t+ctcl 14167 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-ral 3108 df-rex 3109 df-rab 3112 df-v 3434 df-sbc 3702 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-int 4777 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-iota 6181 df-fun 6219 df-fv 6225 df-trcl 14169 |
This theorem is referenced by: brcnvtrclfv 14185 brtrclfvcnv 14186 trclfvcotr 14191 |
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