Mathbox for Zhi Wang |
< Previous
Next >
Nearby theorems |
||
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 2739 | . . . 4 ⊢ (SetCat‘𝑈) = (SetCat‘𝑈) | |
3 | setcthin.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | 2, 3 | setcbas 17462 | . . 3 ⊢ (𝜑 → 𝑈 = (Base‘(SetCat‘𝑈))) |
5 | eqidd 2740 | . . 3 ⊢ (𝜑 → (Hom ‘(SetCat‘𝑈)) = (Hom ‘(SetCat‘𝑈))) | |
6 | elequ2 2129 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑝 ∈ 𝑥 ↔ 𝑝 ∈ 𝑧)) | |
7 | 6 | mobidv 2550 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (∃*𝑝 𝑝 ∈ 𝑥 ↔ ∃*𝑝 𝑝 ∈ 𝑧)) |
8 | setcthin.x | . . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 ∃*𝑝 𝑝 ∈ 𝑥) | |
9 | 8 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → ∀𝑥 ∈ 𝑈 ∃*𝑝 𝑝 ∈ 𝑥) |
10 | simprr 773 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑧 ∈ 𝑈) | |
11 | 7, 9, 10 | rspcdva 3531 | . . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → ∃*𝑝 𝑝 ∈ 𝑧) |
12 | mofmo 45751 | . . . . 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 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑈 ∈ 𝑉) |
15 | eqid 2739 | . . . . . 6 ⊢ (Hom ‘(SetCat‘𝑈)) = (Hom ‘(SetCat‘𝑈)) | |
16 | simprl 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → 𝑦 ∈ 𝑈) | |
17 | 2, 14, 15, 16, 10 | elsetchom 17465 | . . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → (𝑓 ∈ (𝑦(Hom ‘(SetCat‘𝑈))𝑧) ↔ 𝑓:𝑦⟶𝑧)) |
18 | 17 | mobidv 2550 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → (∃*𝑓 𝑓 ∈ (𝑦(Hom ‘(SetCat‘𝑈))𝑧) ↔ ∃*𝑓 𝑓:𝑦⟶𝑧)) |
19 | 13, 18 | mpbird 260 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑈 ∧ 𝑧 ∈ 𝑈)) → ∃*𝑓 𝑓 ∈ (𝑦(Hom ‘(SetCat‘𝑈))𝑧)) |
20 | 2 | setccat 17469 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → (SetCat‘𝑈) ∈ Cat) |
21 | 3, 20 | syl 17 | . . 3 ⊢ (𝜑 → (SetCat‘𝑈) ∈ Cat) |
22 | 4, 5, 19, 21 | isthincd 45838 | . 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 399 = wceq 1542 ∈ wcel 2114 ∃*wmo 2539 ∀wral 3054 ⟶wf 6345 ‘cfv 6349 (class class class)co 7182 Hom chom 16691 Catccat 17050 SetCatcsetc 17459 ThinCatcthinc 45826 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-map 8451 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-7 11796 df-8 11797 df-9 11798 df-n0 11989 df-z 12075 df-dec 12192 df-uz 12337 df-fz 12994 df-struct 16600 df-ndx 16601 df-slot 16602 df-base 16604 df-hom 16704 df-cco 16705 df-cat 17054 df-cid 17055 df-setc 17460 df-thinc 45827 |
This theorem is referenced by: setc2othin 45848 |
Copyright terms: Public domain | W3C validator |