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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 2771 | . . . 4 ⊢ (Sect‘𝐶) = (Sect‘𝐶) | |
9 | eqid 2771 | . . . 4 ⊢ (Sect‘𝐷) = (Sect‘𝐷) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | fthsect 16792 | . . 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 16792 | . . 3 ⊢ (𝜑 → (𝑁(𝑌(Sect‘𝐶)𝑋)𝑀 ↔ ((𝑌𝐺𝑋)‘𝑁)((𝐹‘𝑌)(Sect‘𝐷)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀))) |
12 | 10, 11 | anbi12d 616 | . 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 16774 | . . . . . . . 8 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷) | |
15 | 14 | ssbri 4832 | . . . . . . 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 4788 | . . . . . 6 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) | |
18 | 16, 17 | sylib 208 | . . . . 5 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) |
19 | funcrcl 16730 | . . . . 5 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) | |
20 | 18, 19 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) |
21 | 20 | simpld 482 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
22 | 1, 13, 21, 4, 5, 8 | isinv 16627 | . 2 ⊢ (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐶)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐶)𝑋)𝑀))) |
23 | eqid 2771 | . . 3 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
24 | fthinv.t | . . 3 ⊢ 𝐽 = (Inv‘𝐷) | |
25 | 20 | simprd 483 | . . 3 ⊢ (𝜑 → 𝐷 ∈ Cat) |
26 | 1, 23, 16 | funcf1 16733 | . . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐷)) |
27 | 26, 4 | ffvelrnd 6505 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐷)) |
28 | 26, 5 | ffvelrnd 6505 | . . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐷)) |
29 | 23, 24, 25, 27, 28, 9 | isinv 16627 | . 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 300 | 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 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 〈cop 4323 class class class wbr 4787 ‘cfv 6030 (class class class)co 6796 Basecbs 16064 Hom chom 16160 Catccat 16532 Sectcsect 16611 Invcinv 16612 Func cfunc 16721 Faith cfth 16770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-1st 7319 df-2nd 7320 df-map 8015 df-ixp 8067 df-cat 16536 df-cid 16537 df-sect 16614 df-inv 16615 df-func 16725 df-fth 16772 |
This theorem is referenced by: ffthiso 16796 |
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