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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 2737 | . . . 4 ⊢ (Sect‘𝐶) = (Sect‘𝐶) | |
9 | eqid 2737 | . . . 4 ⊢ (Sect‘𝐷) = (Sect‘𝐷) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | fthsect 17432 | . . 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 17432 | . . 3 ⊢ (𝜑 → (𝑁(𝑌(Sect‘𝐶)𝑋)𝑀 ↔ ((𝑌𝐺𝑋)‘𝑁)((𝐹‘𝑌)(Sect‘𝐷)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀))) |
12 | 10, 11 | anbi12d 634 | . 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 17414 | . . . . . . . 8 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷) | |
15 | 14 | ssbri 5098 | . . . . . . 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 5054 | . . . . . 6 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) | |
18 | 16, 17 | sylib 221 | . . . . 5 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) |
19 | funcrcl 17369 | . . . . 5 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) | |
20 | 18, 19 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) |
21 | 20 | simpld 498 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
22 | 1, 13, 21, 4, 5, 8 | isinv 17265 | . 2 ⊢ (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐶)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐶)𝑋)𝑀))) |
23 | eqid 2737 | . . 3 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
24 | fthinv.t | . . 3 ⊢ 𝐽 = (Inv‘𝐷) | |
25 | 20 | simprd 499 | . . 3 ⊢ (𝜑 → 𝐷 ∈ Cat) |
26 | 1, 23, 16 | funcf1 17372 | . . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐷)) |
27 | 26, 4 | ffvelrnd 6905 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐷)) |
28 | 26, 5 | ffvelrnd 6905 | . . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐷)) |
29 | 23, 24, 25, 27, 28, 9 | isinv 17265 | . 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 314 | 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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 〈cop 4547 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 Hom chom 16813 Catccat 17167 Sectcsect 17249 Invcinv 17250 Func cfunc 17360 Faith cfth 17410 |
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 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-map 8510 df-ixp 8579 df-cat 17171 df-cid 17172 df-sect 17252 df-inv 17253 df-func 17364 df-fth 17412 |
This theorem is referenced by: ffthiso 17436 |
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