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Mirrors > Home > MPE Home > Th. List > sscrel | Structured version Visualization version GIF version |
Description: The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
sscrel | ⊢ Rel ⊆cat |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ssc 17079 | . 2 ⊢ ⊆cat = {〈ℎ, 𝑗〉 ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))} | |
2 | 1 | relopabi 5693 | 1 ⊢ Rel ⊆cat |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∃wex 1776 ∈ wcel 2110 ∃wrex 3139 𝒫 cpw 4538 × cxp 5552 Rel wrel 5559 Fn wfn 6349 ‘cfv 6354 Xcixp 8460 ⊆cat cssc 17076 |
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 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-opab 5128 df-xp 5560 df-rel 5561 df-ssc 17079 |
This theorem is referenced by: brssc 17083 ssc1 17090 ssc2 17091 ssctr 17094 issubc 17104 |
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