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Mirrors > Home > MPE Home > Th. List > trclfvlb | Structured version Visualization version GIF version |
Description: The transitive closure of a relation has a lower bound. (Contributed by RP, 28-Apr-2020.) |
Ref | Expression |
---|---|
trclfvlb | ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (t+‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssmin 4431 | . 2 ⊢ 𝑅 ⊆ ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} | |
2 | trclfv 13589 | . 2 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}) | |
3 | 1, 2 | syl5sseqr 3617 | 1 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (t+‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 {cab 2596 ⊆ wss 3540 ∩ cint 4410 ∘ ccom 5042 ‘cfv 5804 t+ctcl 13572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-iota 5768 df-fun 5806 df-fv 5812 df-trcl 13574 |
This theorem is referenced by: trclfvlb2 13599 trclfvlb3 13600 cotrtrclfv 13601 trclfvg 13604 dmtrclfv 13607 rntrclfvOAI 36272 brtrclfv2 37038 frege96d 37060 frege91d 37062 frege97d 37063 frege109d 37068 frege131d 37075 |
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