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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 4969 | . 2 ⊢ 𝑅 ⊆ ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} | |
2 | trclfv 14942 | . 2 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}) | |
3 | 1, 2 | sseqtrrid 4033 | 1 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (t+‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 {cab 2710 ⊆ wss 3946 ∩ cint 4948 ∘ ccom 5678 ‘cfv 6539 t+ctcl 14927 |
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 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 |
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 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-int 4949 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-iota 6491 df-fun 6541 df-fv 6547 df-trcl 14929 |
This theorem is referenced by: trclfvlb2 14952 trclfvlb3 14953 cotrtrclfv 14954 trclfvg 14957 dmtrclfv 14960 rntrclfvOAI 41361 brtrclfv2 42410 frege96d 42432 frege91d 42434 frege97d 42435 frege109d 42440 frege131d 42447 |
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