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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 17819 | . 2 ⊢ ⊆cat = {〈ℎ, 𝑗〉 ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))} | |
2 | 1 | relopabiv 5817 | 1 ⊢ Rel ⊆cat |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∃wex 1774 ∈ wcel 2099 ∃wrex 3060 𝒫 cpw 4598 × cxp 5671 Rel wrel 5678 Fn wfn 6539 ‘cfv 6544 Xcixp 8916 ⊆cat cssc 17816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-v 3465 df-ss 3964 df-opab 5207 df-xp 5679 df-rel 5680 df-ssc 17819 |
This theorem is referenced by: brssc 17823 ssc1 17830 ssc2 17831 ssctr 17834 issubc 17847 |
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