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Mirrors > Home > MPE Home > Th. List > fthinv | Structured version Visualization version GIF version |
Description: A faithful functor reflects inverses. (Contributed by Mario Carneiro, 27-Jan-2017.) |
Ref | Expression |
---|---|
fthsect.b | ⊢ 𝐵 = (Base‘𝐶) |
fthsect.h | ⊢ 𝐻 = (Hom ‘𝐶) |
fthsect.f | ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺) |
fthsect.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
fthsect.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
fthsect.m | ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌)) |
fthsect.n | ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐻𝑋)) |
fthinv.s | ⊢ 𝐼 = (Inv‘𝐶) |
fthinv.t | ⊢ 𝐽 = (Inv‘𝐷) |
Ref | Expression |
---|---|
fthinv | ⊢ (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝐽(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fthsect.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
2 | fthsect.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐶) | |
3 | fthsect.f | . . . 4 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺) | |
4 | fthsect.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | fthsect.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | fthsect.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌)) | |
7 | fthsect.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐻𝑋)) | |
8 | eqid 2797 | . . . 4 ⊢ (Sect‘𝐶) = (Sect‘𝐶) | |
9 | eqid 2797 | . . . 4 ⊢ (Sect‘𝐷) = (Sect‘𝐷) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | fthsect 17028 | . . 3 ⊢ (𝜑 → (𝑀(𝑋(Sect‘𝐶)𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)(Sect‘𝐷)(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁))) |
11 | 1, 2, 3, 5, 4, 7, 6, 8, 9 | fthsect 17028 | . . 3 ⊢ (𝜑 → (𝑁(𝑌(Sect‘𝐶)𝑋)𝑀 ↔ ((𝑌𝐺𝑋)‘𝑁)((𝐹‘𝑌)(Sect‘𝐷)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀))) |
12 | 10, 11 | anbi12d 630 | . 2 ⊢ (𝜑 → ((𝑀(𝑋(Sect‘𝐶)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐶)𝑋)𝑀) ↔ (((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)(Sect‘𝐷)(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁) ∧ ((𝑌𝐺𝑋)‘𝑁)((𝐹‘𝑌)(Sect‘𝐷)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)))) |
13 | fthinv.s | . . 3 ⊢ 𝐼 = (Inv‘𝐶) | |
14 | fthfunc 17010 | . . . . . . . 8 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷) | |
15 | 14 | ssbri 5013 | . . . . . . 7 ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺) |
16 | 3, 15 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) |
17 | df-br 4969 | . . . . . 6 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) | |
18 | 16, 17 | sylib 219 | . . . . 5 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) |
19 | funcrcl 16966 | . . . . 5 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) | |
20 | 18, 19 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) |
21 | 20 | simpld 495 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
22 | 1, 13, 21, 4, 5, 8 | isinv 16863 | . 2 ⊢ (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐶)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐶)𝑋)𝑀))) |
23 | eqid 2797 | . . 3 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
24 | fthinv.t | . . 3 ⊢ 𝐽 = (Inv‘𝐷) | |
25 | 20 | simprd 496 | . . 3 ⊢ (𝜑 → 𝐷 ∈ Cat) |
26 | 1, 23, 16 | funcf1 16969 | . . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐷)) |
27 | 26, 4 | ffvelrnd 6724 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐷)) |
28 | 26, 5 | ffvelrnd 6724 | . . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐷)) |
29 | 23, 24, 25, 27, 28, 9 | isinv 16863 | . 2 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝐽(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁) ↔ (((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)(Sect‘𝐷)(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁) ∧ ((𝑌𝐺𝑋)‘𝑁)((𝐹‘𝑌)(Sect‘𝐷)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)))) |
30 | 12, 22, 29 | 3bitr4d 312 | 1 ⊢ (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝐽(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1525 ∈ wcel 2083 〈cop 4484 class class class wbr 4968 ‘cfv 6232 (class class class)co 7023 Basecbs 16316 Hom chom 16409 Catccat 16768 Sectcsect 16847 Invcinv 16848 Func cfunc 16957 Faith cfth 17006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-1st 7552 df-2nd 7553 df-map 8265 df-ixp 8318 df-cat 16772 df-cid 16773 df-sect 16850 df-inv 16851 df-func 16961 df-fth 17008 |
This theorem is referenced by: ffthiso 17032 |
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