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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 2737 | . . 3 ⊢ (Sect‘𝐷) = (Sect‘𝐷) | |
3 | eqid 2737 | . . 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 5104 | . . . . . . . . 9 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) | |
10 | 4, 9 | sylib 217 | . . . . . . . 8 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) |
11 | funcrcl 17709 | . . . . . . . 8 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) | |
12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) |
13 | 12 | simpld 495 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat) |
14 | 1, 8, 13, 5, 6, 2 | isinv 17603 | . . . . 5 ⊢ (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐷)𝑋)𝑀))) |
15 | 7, 14 | mpbid 231 | . . . 4 ⊢ (𝜑 → (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐷)𝑋)𝑀)) |
16 | 15 | simpld 495 | . . 3 ⊢ (𝜑 → 𝑀(𝑋(Sect‘𝐷)𝑌)𝑁) |
17 | 1, 2, 3, 4, 5, 6, 16 | funcsect 17718 | . 2 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)(Sect‘𝐸)(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁)) |
18 | 15 | simprd 496 | . . 3 ⊢ (𝜑 → 𝑁(𝑌(Sect‘𝐷)𝑋)𝑀) |
19 | 1, 2, 3, 4, 6, 5, 18 | funcsect 17718 | . 2 ⊢ (𝜑 → ((𝑌𝐺𝑋)‘𝑁)((𝐹‘𝑌)(Sect‘𝐸)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)) |
20 | eqid 2737 | . . 3 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
21 | funcinv.t | . . 3 ⊢ 𝐽 = (Inv‘𝐸) | |
22 | 12 | simprd 496 | . . 3 ⊢ (𝜑 → 𝐸 ∈ Cat) |
23 | 1, 20, 4 | funcf1 17712 | . . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐸)) |
24 | 23, 5 | ffvelcdmd 7032 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐸)) |
25 | 23, 6 | ffvelcdmd 7032 | . . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐸)) |
26 | 20, 21, 22, 24, 25, 3 | isinv 17603 | . 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 396 = wceq 1541 ∈ wcel 2106 〈cop 4590 class class class wbr 5103 ‘cfv 6493 (class class class)co 7351 Basecbs 17043 Catccat 17504 Sectcsect 17587 Invcinv 17588 Func cfunc 17700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-map 8725 df-ixp 8794 df-sect 17590 df-inv 17591 df-func 17704 |
This theorem is referenced by: funciso 17720 |
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