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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 16859 | . 2 ⊢ ⊆cat = {〈ℎ, 𝑗〉 ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))} | |
2 | 1 | relopabi 5493 | 1 ⊢ Rel ⊆cat |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 ∃wex 1823 ∈ wcel 2107 ∃wrex 3091 𝒫 cpw 4379 × cxp 5355 Rel wrel 5362 Fn wfn 6132 ‘cfv 6137 Xcixp 8196 ⊆cat cssc 16856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-opab 4951 df-xp 5363 df-rel 5364 df-ssc 16859 |
This theorem is referenced by: brssc 16863 ssc1 16870 ssc2 16871 ssctr 16874 issubc 16884 |
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