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Mirrors > Home > MPE Home > Th. List > Mathboxes > postc | Structured version Visualization version GIF version |
Description: The converted category is a poset iff no distinct objects are isomorphic. (Contributed by Zhi Wang, 25-Sep-2024.) |
Ref | Expression |
---|---|
postc.c | ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) |
postc.k | ⊢ (𝜑 → 𝐾 ∈ Proset ) |
postc.b | ⊢ 𝐵 = (Base‘𝐶) |
Ref | Expression |
---|---|
postc | ⊢ (𝜑 → (𝐶 ∈ Poset ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥( ≃𝑐 ‘𝐶)𝑦 → 𝑥 = 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | postc.c | . . . 4 ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) | |
2 | postc.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Proset ) | |
3 | 1, 2 | prstcprs 45970 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Proset ) |
4 | postc.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
5 | eqid 2736 | . . . . 5 ⊢ (le‘𝐶) = (le‘𝐶) | |
6 | 4, 5 | ispos2 17776 | . . . 4 ⊢ (𝐶 ∈ Poset ↔ (𝐶 ∈ Proset ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(le‘𝐶)𝑦 ∧ 𝑦(le‘𝐶)𝑥) → 𝑥 = 𝑦))) |
7 | 6 | baib 539 | . . 3 ⊢ (𝐶 ∈ Proset → (𝐶 ∈ Poset ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(le‘𝐶)𝑦 ∧ 𝑦(le‘𝐶)𝑥) → 𝑥 = 𝑦))) |
8 | 3, 7 | syl 17 | . 2 ⊢ (𝜑 → (𝐶 ∈ Poset ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(le‘𝐶)𝑦 ∧ 𝑦(le‘𝐶)𝑥) → 𝑥 = 𝑦))) |
9 | 1 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 = (ProsetToCat‘𝐾)) |
10 | 2 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐾 ∈ Proset ) |
11 | 9, 10 | prstcthin 45971 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ ThinCat) |
12 | simprl 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵) | |
13 | simprr 773 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵) | |
14 | eqid 2736 | . . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
15 | 11, 4, 12, 13, 14 | thinccic 45958 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ≃𝑐 ‘𝐶)𝑦 ↔ ((𝑥(Hom ‘𝐶)𝑦) ≠ ∅ ∧ (𝑦(Hom ‘𝐶)𝑥) ≠ ∅))) |
16 | eqidd 2737 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (le‘𝐶) = (le‘𝐶)) | |
17 | eqidd 2737 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (Hom ‘𝐶) = (Hom ‘𝐶)) | |
18 | 12, 4 | eleqtrdi 2841 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐶)) |
19 | 13, 4 | eleqtrdi 2841 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (Base‘𝐶)) |
20 | 9, 10, 16, 17, 18, 19 | prstchom 45972 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(le‘𝐶)𝑦 ↔ (𝑥(Hom ‘𝐶)𝑦) ≠ ∅)) |
21 | 9, 10, 16, 17, 19, 18 | prstchom 45972 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦(le‘𝐶)𝑥 ↔ (𝑦(Hom ‘𝐶)𝑥) ≠ ∅)) |
22 | 20, 21 | anbi12d 634 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(le‘𝐶)𝑦 ∧ 𝑦(le‘𝐶)𝑥) ↔ ((𝑥(Hom ‘𝐶)𝑦) ≠ ∅ ∧ (𝑦(Hom ‘𝐶)𝑥) ≠ ∅))) |
23 | 15, 22 | bitr4d 285 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ≃𝑐 ‘𝐶)𝑦 ↔ (𝑥(le‘𝐶)𝑦 ∧ 𝑦(le‘𝐶)𝑥))) |
24 | 23 | imbi1d 345 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥( ≃𝑐 ‘𝐶)𝑦 → 𝑥 = 𝑦) ↔ ((𝑥(le‘𝐶)𝑦 ∧ 𝑦(le‘𝐶)𝑥) → 𝑥 = 𝑦))) |
25 | 24 | 2ralbidva 3109 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥( ≃𝑐 ‘𝐶)𝑦 → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(le‘𝐶)𝑦 ∧ 𝑦(le‘𝐶)𝑥) → 𝑥 = 𝑦))) |
26 | 8, 25 | bitr4d 285 | 1 ⊢ (𝜑 → (𝐶 ∈ Poset ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥( ≃𝑐 ‘𝐶)𝑦 → 𝑥 = 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∀wral 3051 ∅c0 4223 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 lecple 16756 Hom chom 16760 ≃𝑐 ccic 17254 Proset cproset 17754 Posetcpo 17768 ProsetToCatcprstc 45959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ple 16769 df-hom 16773 df-cco 16774 df-cat 17125 df-cid 17126 df-sect 17206 df-inv 17207 df-iso 17208 df-cic 17255 df-proset 17756 df-poset 17774 df-thinc 45917 df-prstc 45960 |
This theorem is referenced by: (None) |
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