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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 2824 | . . . 4 ⊢ (Sect‘𝐶) = (Sect‘𝐶) | |
9 | eqid 2824 | . . . 4 ⊢ (Sect‘𝐷) = (Sect‘𝐷) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | fthsect 17198 | . . 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 17198 | . . 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 17180 | . . . . . . . 8 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷) | |
15 | 14 | ssbri 5114 | . . . . . . 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 5070 | . . . . . 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 17136 | . . . . 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 17033 | . 2 ⊢ (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐶)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐶)𝑋)𝑀))) |
23 | eqid 2824 | . . 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 17139 | . . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐷)) |
27 | 26, 4 | ffvelrnd 6855 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐷)) |
28 | 26, 5 | ffvelrnd 6855 | . . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐷)) |
29 | 23, 24, 25, 27, 28, 9 | isinv 17033 | . 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 1536 ∈ wcel 2113 〈cop 4576 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 Hom chom 16579 Catccat 16938 Sectcsect 17017 Invcinv 17018 Func cfunc 17127 Faith cfth 17176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-map 8411 df-ixp 8465 df-cat 16942 df-cid 16943 df-sect 17020 df-inv 17021 df-func 17131 df-fth 17178 |
This theorem is referenced by: ffthiso 17202 |
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