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Mirrors > Home > MPE Home > Th. List > brcnvtrclfv | Structured version Visualization version GIF version |
Description: Two ways of expressing the transitive closure of the converse of a binary relation. (Contributed by RP, 9-May-2020.) |
Ref | Expression |
---|---|
brcnvtrclfv | ⊢ ((𝑅 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴◡(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐵𝑟𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brcnvg 5839 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴◡(t+‘𝑅)𝐵 ↔ 𝐵(t+‘𝑅)𝐴)) | |
2 | 1 | 3adant1 1131 | . 2 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴◡(t+‘𝑅)𝐵 ↔ 𝐵(t+‘𝑅)𝐴)) |
3 | brtrclfv 14896 | . . 3 ⊢ (𝑅 ∈ 𝑈 → (𝐵(t+‘𝑅)𝐴 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐵𝑟𝐴))) | |
4 | 3 | 3ad2ant1 1134 | . 2 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵(t+‘𝑅)𝐴 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐵𝑟𝐴))) |
5 | 2, 4 | bitrd 279 | 1 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴◡(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐵𝑟𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∀wal 1540 ∈ wcel 2107 ⊆ wss 3914 class class class wbr 5109 ◡ccnv 5636 ∘ ccom 5641 ‘cfv 6500 t+ctcl 14879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-iota 6452 df-fun 6502 df-fv 6508 df-trcl 14881 |
This theorem is referenced by: brcnvtrclfvcnv 14899 |
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