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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 2823 | . 2 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
2 | eqid 2823 | . 2 ⊢ (Inv‘𝐸) = (Inv‘𝐸) | |
3 | funciso.f | . . . . 5 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) | |
4 | df-br 5069 | . . . . 5 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) | |
5 | 3, 4 | sylib 220 | . . . 4 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) |
6 | funcrcl 17135 | . . . 4 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) |
8 | 7 | simprd 498 | . 2 ⊢ (𝜑 → 𝐸 ∈ Cat) |
9 | funciso.b | . . . 4 ⊢ 𝐵 = (Base‘𝐷) | |
10 | 9, 1, 3 | funcf1 17138 | . . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐸)) |
11 | funciso.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
12 | 10, 11 | ffvelrnd 6854 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐸)) |
13 | funciso.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
14 | 10, 13 | ffvelrnd 6854 | . 2 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐸)) |
15 | funciso.t | . 2 ⊢ 𝐽 = (Iso‘𝐸) | |
16 | eqid 2823 | . . 3 ⊢ (Inv‘𝐷) = (Inv‘𝐷) | |
17 | funciso.s | . . . 4 ⊢ 𝐼 = (Iso‘𝐷) | |
18 | 7 | simpld 497 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat) |
19 | funciso.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐼𝑌)) | |
20 | 9, 17, 16, 18, 11, 13, 19 | invisoinvr 17063 | . . 3 ⊢ (𝜑 → 𝑀(𝑋(Inv‘𝐷)𝑌)((𝑋(Inv‘𝐷)𝑌)‘𝑀)) |
21 | 9, 16, 2, 3, 11, 13, 20 | funcinv 17145 | . 2 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)(Inv‘𝐸)(𝐹‘𝑌))((𝑌𝐺𝑋)‘((𝑋(Inv‘𝐷)𝑌)‘𝑀))) |
22 | 1, 2, 8, 12, 14, 15, 21 | inviso1 17038 | 1 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 〈cop 4575 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 Catccat 16937 Invcinv 17017 Isociso 17018 Func cfunc 17126 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 df-ixp 8464 df-cat 16941 df-cid 16942 df-sect 17019 df-inv 17020 df-iso 17021 df-func 17130 |
This theorem is referenced by: ffthiso 17201 |
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