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Mirrors > Home > MPE Home > Th. List > Mathboxes > subthinc | Structured version Visualization version GIF version |
Description: A subcategory of a thin category is thin. (Contributed by Zhi Wang, 30-Sep-2024.) |
Ref | Expression |
---|---|
subthinc.1 | ⊢ 𝐷 = (𝐶 ↾cat 𝐽) |
subthinc.j | ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) |
subthinc.c | ⊢ (𝜑 → 𝐶 ∈ ThinCat) |
Ref | Expression |
---|---|
subthinc | ⊢ (𝜑 → 𝐷 ∈ ThinCat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subthinc.1 | . . 3 ⊢ 𝐷 = (𝐶 ↾cat 𝐽) | |
2 | eqid 2735 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | subthinc.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ThinCat) | |
4 | subthinc.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) | |
5 | eqidd 2736 | . . . 4 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽) | |
6 | 4, 5 | subcfn 17892 | . . 3 ⊢ (𝜑 → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽)) |
7 | 4, 6, 2 | subcss1 17893 | . . 3 ⊢ (𝜑 → dom dom 𝐽 ⊆ (Base‘𝐶)) |
8 | 1, 2, 3, 6, 7 | rescbas 17877 | . 2 ⊢ (𝜑 → dom dom 𝐽 = (Base‘𝐷)) |
9 | 1, 2, 3, 6, 7 | reschom 17879 | . 2 ⊢ (𝜑 → 𝐽 = (Hom ‘𝐷)) |
10 | 4 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → 𝐽 ∈ (Subcat‘𝐶)) |
11 | 6 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽)) |
12 | eqid 2735 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
13 | simprl 771 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → 𝑥 ∈ dom dom 𝐽) | |
14 | simprr 773 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → 𝑦 ∈ dom dom 𝐽) | |
15 | 10, 11, 12, 13, 14 | subcss2 17894 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → (𝑥𝐽𝑦) ⊆ (𝑥(Hom ‘𝐶)𝑦)) |
16 | 3 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → 𝐶 ∈ ThinCat) |
17 | 7 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → dom dom 𝐽 ⊆ (Base‘𝐶)) |
18 | 17, 13 | sseldd 3996 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → 𝑥 ∈ (Base‘𝐶)) |
19 | 17, 14 | sseldd 3996 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → 𝑦 ∈ (Base‘𝐶)) |
20 | 16, 18, 19, 2, 12 | thincmo 48829 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) |
21 | mosssn2 48665 | . . . . 5 ⊢ (∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ ∃𝑓(𝑥(Hom ‘𝐶)𝑦) ⊆ {𝑓}) | |
22 | 20, 21 | sylib 218 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → ∃𝑓(𝑥(Hom ‘𝐶)𝑦) ⊆ {𝑓}) |
23 | sstr2 4002 | . . . . 5 ⊢ ((𝑥𝐽𝑦) ⊆ (𝑥(Hom ‘𝐶)𝑦) → ((𝑥(Hom ‘𝐶)𝑦) ⊆ {𝑓} → (𝑥𝐽𝑦) ⊆ {𝑓})) | |
24 | 23 | eximdv 1915 | . . . 4 ⊢ ((𝑥𝐽𝑦) ⊆ (𝑥(Hom ‘𝐶)𝑦) → (∃𝑓(𝑥(Hom ‘𝐶)𝑦) ⊆ {𝑓} → ∃𝑓(𝑥𝐽𝑦) ⊆ {𝑓})) |
25 | 15, 22, 24 | sylc 65 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → ∃𝑓(𝑥𝐽𝑦) ⊆ {𝑓}) |
26 | mosssn2 48665 | . . 3 ⊢ (∃*𝑓 𝑓 ∈ (𝑥𝐽𝑦) ↔ ∃𝑓(𝑥𝐽𝑦) ⊆ {𝑓}) | |
27 | 25, 26 | sylibr 234 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → ∃*𝑓 𝑓 ∈ (𝑥𝐽𝑦)) |
28 | 1, 4 | subccat 17899 | . 2 ⊢ (𝜑 → 𝐷 ∈ Cat) |
29 | 8, 9, 27, 28 | isthincd 48837 | 1 ⊢ (𝜑 → 𝐷 ∈ ThinCat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ∃*wmo 2536 ⊆ wss 3963 {csn 4631 × cxp 5687 dom cdm 5689 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Hom chom 17309 ↾cat cresc 17856 Subcatcsubc 17857 ThinCatcthinc 48819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-hom 17322 df-cco 17323 df-cat 17713 df-cid 17714 df-homf 17715 df-ssc 17858 df-resc 17859 df-subc 17860 df-thinc 48820 |
This theorem is referenced by: (None) |
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