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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 17269 | . 2 ⊢ ⊆cat = {〈ℎ, 𝑗〉 ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))} | |
2 | 1 | relopabiv 5675 | 1 ⊢ Rel ⊆cat |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∃wex 1787 ∈ wcel 2112 ∃wrex 3052 𝒫 cpw 4499 × cxp 5534 Rel wrel 5541 Fn wfn 6353 ‘cfv 6358 Xcixp 8556 ⊆cat cssc 17266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 df-in 3860 df-ss 3870 df-opab 5102 df-xp 5542 df-rel 5543 df-ssc 17269 |
This theorem is referenced by: brssc 17273 ssc1 17280 ssc2 17281 ssctr 17284 issubc 17295 |
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