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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 2798 | . . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
3 | subcss1.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
4 | 2, 3 | homffn 16955 | . . 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 17102 | . 2 ⊢ (𝜑 → 𝐽 ⊆cat (Homf ‘𝐶)) |
8 | 1, 5, 7 | ssc1 17083 | 1 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 × cxp 5517 Fn wfn 6319 ‘cfv 6324 Basecbs 16475 Homf chomf 16929 Subcatcsubc 17071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-pm 8392 df-ixp 8445 df-homf 16933 df-ssc 17072 df-subc 17074 |
This theorem is referenced by: subcss2 17105 subccatid 17108 subsubc 17115 funcres 17158 funcres2b 17159 funcres2 17160 idfusubc 44490 |
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