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Mirrors > Home > MPE Home > Th. List > funcinv | Structured version Visualization version GIF version |
Description: The image of an inverse under a functor is an inverse. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
funcinv.b | ⊢ 𝐵 = (Base‘𝐷) |
funcinv.s | ⊢ 𝐼 = (Inv‘𝐷) |
funcinv.t | ⊢ 𝐽 = (Inv‘𝐸) |
funcinv.f | ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) |
funcinv.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
funcinv.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
funcinv.m | ⊢ (𝜑 → 𝑀(𝑋𝐼𝑌)𝑁) |
Ref | Expression |
---|---|
funcinv | ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝐽(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcinv.b | . . 3 ⊢ 𝐵 = (Base‘𝐷) | |
2 | eqid 2821 | . . 3 ⊢ (Sect‘𝐷) = (Sect‘𝐷) | |
3 | eqid 2821 | . . 3 ⊢ (Sect‘𝐸) = (Sect‘𝐸) | |
4 | funcinv.f | . . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) | |
5 | funcinv.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | funcinv.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | funcinv.m | . . . . 5 ⊢ (𝜑 → 𝑀(𝑋𝐼𝑌)𝑁) | |
8 | funcinv.s | . . . . . 6 ⊢ 𝐼 = (Inv‘𝐷) | |
9 | df-br 5059 | . . . . . . . . 9 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) | |
10 | 4, 9 | sylib 220 | . . . . . . . 8 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) |
11 | funcrcl 17127 | . . . . . . . 8 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) | |
12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) |
13 | 12 | simpld 497 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat) |
14 | 1, 8, 13, 5, 6, 2 | isinv 17024 | . . . . 5 ⊢ (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐷)𝑋)𝑀))) |
15 | 7, 14 | mpbid 234 | . . . 4 ⊢ (𝜑 → (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐷)𝑋)𝑀)) |
16 | 15 | simpld 497 | . . 3 ⊢ (𝜑 → 𝑀(𝑋(Sect‘𝐷)𝑌)𝑁) |
17 | 1, 2, 3, 4, 5, 6, 16 | funcsect 17136 | . 2 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)(Sect‘𝐸)(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁)) |
18 | 15 | simprd 498 | . . 3 ⊢ (𝜑 → 𝑁(𝑌(Sect‘𝐷)𝑋)𝑀) |
19 | 1, 2, 3, 4, 6, 5, 18 | funcsect 17136 | . 2 ⊢ (𝜑 → ((𝑌𝐺𝑋)‘𝑁)((𝐹‘𝑌)(Sect‘𝐸)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)) |
20 | eqid 2821 | . . 3 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
21 | funcinv.t | . . 3 ⊢ 𝐽 = (Inv‘𝐸) | |
22 | 12 | simprd 498 | . . 3 ⊢ (𝜑 → 𝐸 ∈ Cat) |
23 | 1, 20, 4 | funcf1 17130 | . . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐸)) |
24 | 23, 5 | ffvelrnd 6846 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐸)) |
25 | 23, 6 | ffvelrnd 6846 | . . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐸)) |
26 | 20, 21, 22, 24, 25, 3 | isinv 17024 | . 2 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝐽(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁) ↔ (((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)(Sect‘𝐸)(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁) ∧ ((𝑌𝐺𝑋)‘𝑁)((𝐹‘𝑌)(Sect‘𝐸)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)))) |
27 | 17, 19, 26 | mpbir2and 711 | 1 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝐽(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 〈cop 4566 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Catccat 16929 Sectcsect 17008 Invcinv 17009 Func cfunc 17118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-map 8402 df-ixp 8456 df-sect 17011 df-inv 17012 df-func 17122 |
This theorem is referenced by: funciso 17138 |
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