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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 2735 | . . 3 ⊢ (Sect‘𝐷) = (Sect‘𝐷) | |
3 | eqid 2735 | . . 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 5149 | . . . . . . . . 9 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) | |
10 | 4, 9 | sylib 218 | . . . . . . . 8 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) |
11 | funcrcl 17914 | . . . . . . . 8 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) | |
12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) |
13 | 12 | simpld 494 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat) |
14 | 1, 8, 13, 5, 6, 2 | isinv 17808 | . . . . 5 ⊢ (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐷)𝑋)𝑀))) |
15 | 7, 14 | mpbid 232 | . . . 4 ⊢ (𝜑 → (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐷)𝑋)𝑀)) |
16 | 15 | simpld 494 | . . 3 ⊢ (𝜑 → 𝑀(𝑋(Sect‘𝐷)𝑌)𝑁) |
17 | 1, 2, 3, 4, 5, 6, 16 | funcsect 17923 | . 2 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)(Sect‘𝐸)(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁)) |
18 | 15 | simprd 495 | . . 3 ⊢ (𝜑 → 𝑁(𝑌(Sect‘𝐷)𝑋)𝑀) |
19 | 1, 2, 3, 4, 6, 5, 18 | funcsect 17923 | . 2 ⊢ (𝜑 → ((𝑌𝐺𝑋)‘𝑁)((𝐹‘𝑌)(Sect‘𝐸)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)) |
20 | eqid 2735 | . . 3 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
21 | funcinv.t | . . 3 ⊢ 𝐽 = (Inv‘𝐸) | |
22 | 12 | simprd 495 | . . 3 ⊢ (𝜑 → 𝐸 ∈ Cat) |
23 | 1, 20, 4 | funcf1 17917 | . . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐸)) |
24 | 23, 5 | ffvelcdmd 7105 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐸)) |
25 | 23, 6 | ffvelcdmd 7105 | . . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐸)) |
26 | 20, 21, 22, 24, 25, 3 | isinv 17808 | . 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 713 | 1 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝐽(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 〈cop 4637 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Catccat 17709 Sectcsect 17792 Invcinv 17793 Func cfunc 17905 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-ixp 8937 df-sect 17795 df-inv 17796 df-func 17909 |
This theorem is referenced by: funciso 17925 |
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