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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 16855 | . 2 ⊢ ⊆cat = {〈ℎ, 𝑗〉 ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))} | |
2 | 1 | relopabi 5491 | 1 ⊢ Rel ⊆cat |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 ∃wex 1823 ∈ wcel 2106 ∃wrex 3090 𝒫 cpw 4378 × cxp 5353 Rel wrel 5360 Fn wfn 6130 ‘cfv 6135 Xcixp 8194 ⊆cat cssc 16852 |
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 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 |
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 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-rab 3098 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-opab 4949 df-xp 5361 df-rel 5362 df-ssc 16855 |
This theorem is referenced by: brssc 16859 ssc1 16866 ssc2 16867 ssctr 16870 issubc 16880 |
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