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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 2737 | . . . 4 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
| 3 | subcss1.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
| 4 | 2, 3 | homffn 17736 | . . 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 17885 | . 2 ⊢ (𝜑 → 𝐽 ⊆cat (Homf ‘𝐶)) |
| 8 | 1, 5, 7 | ssc1 17865 | 1 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 × cxp 5683 Fn wfn 6556 ‘cfv 6561 Basecbs 17247 Homf chomf 17709 Subcatcsubc 17853 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-pm 8869 df-ixp 8938 df-homf 17713 df-ssc 17854 df-subc 17856 |
| This theorem is referenced by: subcss2 17888 subccatid 17891 subsubc 17898 funcres 17941 funcres2b 17942 funcres2 17943 idfusubc 17945 subthinc 49092 |
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