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Mirrors > Home > MPE Home > Th. List > subcss1 | Structured version Visualization version GIF version |
Description: The objects of a subcategory are a subset of the objects of the original. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
subcss1.1 | ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) |
subcss1.2 | ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) |
subcss1.b | ⊢ 𝐵 = (Base‘𝐶) |
Ref | Expression |
---|---|
subcss1 | ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subcss1.2 | . 2 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) | |
2 | eqid 2777 | . . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
3 | subcss1.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
4 | 2, 3 | homffn 16738 | . . 3 ⊢ (Homf ‘𝐶) Fn (𝐵 × 𝐵) |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → (Homf ‘𝐶) Fn (𝐵 × 𝐵)) |
6 | subcss1.1 | . . 3 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) | |
7 | 6, 2 | subcssc 16885 | . 2 ⊢ (𝜑 → 𝐽 ⊆cat (Homf ‘𝐶)) |
8 | 1, 5, 7 | ssc1 16866 | 1 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 ⊆ wss 3791 × cxp 5353 Fn wfn 6130 ‘cfv 6135 Basecbs 16255 Homf chomf 16712 Subcatcsubc 16854 |
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-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-pm 8143 df-ixp 8195 df-homf 16716 df-ssc 16855 df-subc 16857 |
This theorem is referenced by: subcss2 16888 subccatid 16891 subsubc 16898 funcres 16941 funcres2b 16942 funcres2 16943 idfusubc 42863 |
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