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Mirrors > Home > MPE Home > Th. List > setcsect | Structured version Visualization version GIF version |
Description: A section in the category of sets, written out. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
setcmon.c | ⊢ 𝐶 = (SetCat‘𝑈) |
setcmon.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
setcmon.x | ⊢ (𝜑 → 𝑋 ∈ 𝑈) |
setcmon.y | ⊢ (𝜑 → 𝑌 ∈ 𝑈) |
setcsect.n | ⊢ 𝑆 = (Sect‘𝐶) |
Ref | Expression |
---|---|
setcsect | ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋 ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2651 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
2 | eqid 2651 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
3 | eqid 2651 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
4 | eqid 2651 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
5 | setcsect.n | . . 3 ⊢ 𝑆 = (Sect‘𝐶) | |
6 | setcmon.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
7 | setcmon.c | . . . . 5 ⊢ 𝐶 = (SetCat‘𝑈) | |
8 | 7 | setccat 16782 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) |
9 | 6, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
10 | setcmon.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑈) | |
11 | 7, 6 | setcbas 16775 | . . . 4 ⊢ (𝜑 → 𝑈 = (Base‘𝐶)) |
12 | 10, 11 | eleqtrd 2732 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
13 | setcmon.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑈) | |
14 | 13, 11 | eleqtrd 2732 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
15 | 1, 2, 3, 4, 5, 9, 12, 14 | issect 16460 | . 2 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))) |
16 | 7, 6, 2, 10, 13 | elsetchom 16778 | . . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹:𝑋⟶𝑌)) |
17 | 7, 6, 2, 13, 10 | elsetchom 16778 | . . . . . 6 ⊢ (𝜑 → (𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ↔ 𝐺:𝑌⟶𝑋)) |
18 | 16, 17 | anbi12d 747 | . . . . 5 ⊢ (𝜑 → ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ↔ (𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋))) |
19 | 18 | anbi1d 741 | . . . 4 ⊢ (𝜑 → (((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ∧ (𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋) ∧ (𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))) |
20 | 6 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋)) → 𝑈 ∈ 𝑉) |
21 | 10 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋)) → 𝑋 ∈ 𝑈) |
22 | 13 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋)) → 𝑌 ∈ 𝑈) |
23 | simprl 809 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋)) → 𝐹:𝑋⟶𝑌) | |
24 | simprr 811 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋)) → 𝐺:𝑌⟶𝑋) | |
25 | 7, 20, 3, 21, 22, 21, 23, 24 | setcco 16780 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋)) → (𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = (𝐺 ∘ 𝐹)) |
26 | 7, 4, 6, 10 | setcid 16783 | . . . . . . 7 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) = ( I ↾ 𝑋)) |
27 | 26 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋)) → ((Id‘𝐶)‘𝑋) = ( I ↾ 𝑋)) |
28 | 25, 27 | eqeq12d 2666 | . . . . 5 ⊢ ((𝜑 ∧ (𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋)) → ((𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋) ↔ (𝐺 ∘ 𝐹) = ( I ↾ 𝑋))) |
29 | 28 | pm5.32da 674 | . . . 4 ⊢ (𝜑 → (((𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋) ∧ (𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝑋)))) |
30 | 19, 29 | bitrd 268 | . . 3 ⊢ (𝜑 → (((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ∧ (𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝑋)))) |
31 | df-3an 1056 | . . 3 ⊢ ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ∧ (𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))) | |
32 | df-3an 1056 | . . 3 ⊢ ((𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋 ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝑋)) ↔ ((𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝑋))) | |
33 | 30, 31, 32 | 3bitr4g 303 | . 2 ⊢ (𝜑 → ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ (𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋 ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝑋)))) |
34 | 15, 33 | bitrd 268 | 1 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹:𝑋⟶𝑌 ∧ 𝐺:𝑌⟶𝑋 ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 〈cop 4216 class class class wbr 4685 I cid 5052 ↾ cres 5145 ∘ ccom 5147 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 Hom chom 15999 compcco 16000 Catccat 16372 Idccid 16373 Sectcsect 16451 SetCatcsetc 16772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-hom 16013 df-cco 16014 df-cat 16376 df-cid 16377 df-sect 16454 df-setc 16773 |
This theorem is referenced by: setcinv 16787 |
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