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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 2821 | . . . 4 ⊢ (Sect‘𝐶) = (Sect‘𝐶) | |
9 | eqid 2821 | . . . 4 ⊢ (Sect‘𝐷) = (Sect‘𝐷) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | fthsect 17195 | . . 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 17195 | . . 3 ⊢ (𝜑 → (𝑁(𝑌(Sect‘𝐶)𝑋)𝑀 ↔ ((𝑌𝐺𝑋)‘𝑁)((𝐹‘𝑌)(Sect‘𝐷)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀))) |
12 | 10, 11 | anbi12d 632 | . 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 17177 | . . . . . . . 8 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷) | |
15 | 14 | ssbri 5111 | . . . . . . 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 5067 | . . . . . 6 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) | |
18 | 16, 17 | sylib 220 | . . . . 5 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) |
19 | funcrcl 17133 | . . . . 5 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) | |
20 | 18, 19 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) |
21 | 20 | simpld 497 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
22 | 1, 13, 21, 4, 5, 8 | isinv 17030 | . 2 ⊢ (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐶)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐶)𝑋)𝑀))) |
23 | eqid 2821 | . . 3 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
24 | fthinv.t | . . 3 ⊢ 𝐽 = (Inv‘𝐷) | |
25 | 20 | simprd 498 | . . 3 ⊢ (𝜑 → 𝐷 ∈ Cat) |
26 | 1, 23, 16 | funcf1 17136 | . . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐷)) |
27 | 26, 4 | ffvelrnd 6852 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐷)) |
28 | 26, 5 | ffvelrnd 6852 | . . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐷)) |
29 | 23, 24, 25, 27, 28, 9 | isinv 17030 | . 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 313 | 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 〈cop 4573 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 Hom chom 16576 Catccat 16935 Sectcsect 17014 Invcinv 17015 Func cfunc 17124 Faith cfth 17173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-map 8408 df-ixp 8462 df-cat 16939 df-cid 16940 df-sect 17017 df-inv 17018 df-func 17128 df-fth 17175 |
This theorem is referenced by: ffthiso 17199 |
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