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Mirrors > Home > MPE Home > Th. List > funciso | Structured version Visualization version GIF version |
Description: The image of an isomorphism under a functor is an isomorphism. Proposition 3.21 of [Adamek] p. 32. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
funciso.b | ⊢ 𝐵 = (Base‘𝐷) |
funciso.s | ⊢ 𝐼 = (Iso‘𝐷) |
funciso.t | ⊢ 𝐽 = (Iso‘𝐸) |
funciso.f | ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) |
funciso.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
funciso.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
funciso.m | ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐼𝑌)) |
Ref | Expression |
---|---|
funciso | ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . 2 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
2 | eqid 2610 | . 2 ⊢ (Inv‘𝐸) = (Inv‘𝐸) | |
3 | funciso.f | . . . . 5 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) | |
4 | df-br 4584 | . . . . 5 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) | |
5 | 3, 4 | sylib 207 | . . . 4 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) |
6 | funcrcl 16346 | . . . 4 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) |
8 | 7 | simprd 478 | . 2 ⊢ (𝜑 → 𝐸 ∈ Cat) |
9 | funciso.b | . . . 4 ⊢ 𝐵 = (Base‘𝐷) | |
10 | 9, 1, 3 | funcf1 16349 | . . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐸)) |
11 | funciso.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
12 | 10, 11 | ffvelrnd 6268 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐸)) |
13 | funciso.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
14 | 10, 13 | ffvelrnd 6268 | . 2 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐸)) |
15 | funciso.t | . 2 ⊢ 𝐽 = (Iso‘𝐸) | |
16 | eqid 2610 | . . 3 ⊢ (Inv‘𝐷) = (Inv‘𝐷) | |
17 | funciso.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐼𝑌)) | |
18 | 7 | simpld 474 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat) |
19 | funciso.s | . . . . . 6 ⊢ 𝐼 = (Iso‘𝐷) | |
20 | 9, 16, 18, 11, 13, 19 | isoval 16248 | . . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐷)𝑌)) |
21 | 17, 20 | eleqtrd 2690 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ dom (𝑋(Inv‘𝐷)𝑌)) |
22 | 9, 16, 18, 11, 13 | invfun 16247 | . . . . 5 ⊢ (𝜑 → Fun (𝑋(Inv‘𝐷)𝑌)) |
23 | funfvbrb 6238 | . . . . 5 ⊢ (Fun (𝑋(Inv‘𝐷)𝑌) → (𝑀 ∈ dom (𝑋(Inv‘𝐷)𝑌) ↔ 𝑀(𝑋(Inv‘𝐷)𝑌)((𝑋(Inv‘𝐷)𝑌)‘𝑀))) | |
24 | 22, 23 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑀 ∈ dom (𝑋(Inv‘𝐷)𝑌) ↔ 𝑀(𝑋(Inv‘𝐷)𝑌)((𝑋(Inv‘𝐷)𝑌)‘𝑀))) |
25 | 21, 24 | mpbid 221 | . . 3 ⊢ (𝜑 → 𝑀(𝑋(Inv‘𝐷)𝑌)((𝑋(Inv‘𝐷)𝑌)‘𝑀)) |
26 | 9, 16, 2, 3, 11, 13, 25 | funcinv 16356 | . 2 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)(Inv‘𝐸)(𝐹‘𝑌))((𝑌𝐺𝑋)‘((𝑋(Inv‘𝐷)𝑌)‘𝑀))) |
27 | 1, 2, 8, 12, 14, 15, 26 | inviso1 16249 | 1 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 〈cop 4131 class class class wbr 4583 dom cdm 5038 Fun wfun 5798 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Catccat 16148 Invcinv 16228 Isociso 16229 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-rmo 2904 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-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 df-ixp 7795 df-cat 16152 df-cid 16153 df-sect 16230 df-inv 16231 df-iso 16232 df-func 16341 |
This theorem is referenced by: ffthiso 16412 |
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