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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 2610 | . . 3 ⊢ (Sect‘𝐷) = (Sect‘𝐷) | |
3 | eqid 2610 | . . 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 4584 | . . . . . . . . 9 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) | |
10 | 4, 9 | sylib 207 | . . . . . . . 8 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) |
11 | funcrcl 16346 | . . . . . . . 8 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) | |
12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) |
13 | 12 | simpld 474 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat) |
14 | 1, 8, 13, 5, 6, 2 | isinv 16243 | . . . . 5 ⊢ (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐷)𝑋)𝑀))) |
15 | 7, 14 | mpbid 221 | . . . 4 ⊢ (𝜑 → (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐷)𝑋)𝑀)) |
16 | 15 | simpld 474 | . . 3 ⊢ (𝜑 → 𝑀(𝑋(Sect‘𝐷)𝑌)𝑁) |
17 | 1, 2, 3, 4, 5, 6, 16 | funcsect 16355 | . 2 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)(Sect‘𝐸)(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁)) |
18 | 15 | simprd 478 | . . 3 ⊢ (𝜑 → 𝑁(𝑌(Sect‘𝐷)𝑋)𝑀) |
19 | 1, 2, 3, 4, 6, 5, 18 | funcsect 16355 | . 2 ⊢ (𝜑 → ((𝑌𝐺𝑋)‘𝑁)((𝐹‘𝑌)(Sect‘𝐸)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)) |
20 | eqid 2610 | . . 3 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
21 | funcinv.t | . . 3 ⊢ 𝐽 = (Inv‘𝐸) | |
22 | 12 | simprd 478 | . . 3 ⊢ (𝜑 → 𝐸 ∈ Cat) |
23 | 1, 20, 4 | funcf1 16349 | . . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐸)) |
24 | 23, 5 | ffvelrnd 6268 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐸)) |
25 | 23, 6 | ffvelrnd 6268 | . . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐸)) |
26 | 20, 21, 22, 24, 25, 3 | isinv 16243 | . 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 959 | 1 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝐽(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 〈cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Catccat 16148 Sectcsect 16227 Invcinv 16228 Func cfunc 16337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 df-ixp 7795 df-sect 16230 df-inv 16231 df-func 16341 |
This theorem is referenced by: funciso 16357 |
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