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Mirrors > Home > MPE Home > Th. List > Mathboxes > setcthin | Structured version Visualization version GIF version |
Description: A category of sets all of whose objects contain at most one element is thin. (Contributed by Zhi Wang, 20-Sep-2024.) |
Ref | Expression |
---|---|
setcthin.c | ⊢ (𝜑 → 𝐶 = (SetCat‘𝑈)) |
setcthin.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
setcthin.x | ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 ∃*𝑝 𝑝 ∈ 𝑥) |
Ref | Expression |
---|---|
setcthin | ⊢ (𝜑 → 𝐶 ∈ ThinCat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setcthin.c | . 2 ⊢ (𝜑 → 𝐶 = (SetCat‘𝑈)) | |
2 | eqid 2733 | . . . 4 ⊢ (SetCat‘𝑈) = (SetCat‘𝑈) | |
3 | setcthin.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | 2, 3 | setcbas 18023 | . . 3 ⊢ (𝜑 → 𝑈 = (Base‘(SetCat‘𝑈))) |
5 | eqidd 2734 | . . 3 ⊢ (𝜑 → (Hom ‘(SetCat‘𝑈)) = (Hom ‘(SetCat‘𝑈))) | |
6 | elequ2 2122 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑝 ∈ 𝑥 ↔ 𝑝 ∈ 𝑧)) | |
7 | 6 | mobidv 2544 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (∃*𝑝 𝑝 ∈ 𝑥 ↔ ∃*𝑝 𝑝 ∈ 𝑧)) |
8 | setcthin.x | . . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 ∃*𝑝 𝑝 ∈ 𝑥) | |
9 | 8 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → ∀𝑥 ∈ 𝑈 ∃*𝑝 𝑝 ∈ 𝑥) |
10 | simprr 772 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑧 ∈ 𝑈) | |
11 | 7, 9, 10 | rspcdva 3612 | . . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → ∃*𝑝 𝑝 ∈ 𝑧) |
12 | mofmo 47414 | . . . . 5 ⊢ (∃*𝑝 𝑝 ∈ 𝑧 → ∃*𝑓 𝑓:𝑦⟶𝑧) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → ∃*𝑓 𝑓:𝑦⟶𝑧) |
14 | 3 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑈 ∈ 𝑉) |
15 | eqid 2733 | . . . . . 6 ⊢ (Hom ‘(SetCat‘𝑈)) = (Hom ‘(SetCat‘𝑈)) | |
16 | simprl 770 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑦 ∈ 𝑈) | |
17 | 2, 14, 15, 16, 10 | elsetchom 18026 | . . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → (𝑓 ∈ (𝑦(Hom ‘(SetCat‘𝑈))𝑧) ↔ 𝑓:𝑦⟶𝑧)) |
18 | 17 | mobidv 2544 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → (∃*𝑓 𝑓 ∈ (𝑦(Hom ‘(SetCat‘𝑈))𝑧) ↔ ∃*𝑓 𝑓:𝑦⟶𝑧)) |
19 | 13, 18 | mpbird 257 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → ∃*𝑓 𝑓 ∈ (𝑦(Hom ‘(SetCat‘𝑈))𝑧)) |
20 | 2 | setccat 18030 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → (SetCat‘𝑈) ∈ Cat) |
21 | 3, 20 | syl 17 | . . 3 ⊢ (𝜑 → (SetCat‘𝑈) ∈ Cat) |
22 | 4, 5, 19, 21 | isthincd 47558 | . 2 ⊢ (𝜑 → (SetCat‘𝑈) ∈ ThinCat) |
23 | 1, 22 | eqeltrd 2834 | 1 ⊢ (𝜑 → 𝐶 ∈ ThinCat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃*wmo 2533 ∀wral 3062 ⟶wf 6535 ‘cfv 6539 (class class class)co 7403 Hom chom 17203 Catccat 17603 SetCatcsetc 18020 ThinCatcthinc 47540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12468 df-z 12554 df-dec 12673 df-uz 12818 df-fz 13480 df-struct 17075 df-slot 17110 df-ndx 17122 df-base 17140 df-hom 17216 df-cco 17217 df-cat 17607 df-cid 17608 df-setc 18021 df-thinc 47541 |
This theorem is referenced by: setc2othin 47577 |
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