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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 2798 | . 2 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
2 | eqid 2798 | . 2 ⊢ (Inv‘𝐸) = (Inv‘𝐸) | |
3 | funciso.f | . . . . 5 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺) | |
4 | df-br 5031 | . . . . 5 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) | |
5 | 3, 4 | sylib 221 | . . . 4 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸)) |
6 | funcrcl 17125 | . . . 4 ⊢ (〈𝐹, 𝐺〉 ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat)) |
8 | 7 | simprd 499 | . 2 ⊢ (𝜑 → 𝐸 ∈ Cat) |
9 | funciso.b | . . . 4 ⊢ 𝐵 = (Base‘𝐷) | |
10 | 9, 1, 3 | funcf1 17128 | . . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐸)) |
11 | funciso.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
12 | 10, 11 | ffvelrnd 6829 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐸)) |
13 | funciso.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
14 | 10, 13 | ffvelrnd 6829 | . 2 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐸)) |
15 | funciso.t | . 2 ⊢ 𝐽 = (Iso‘𝐸) | |
16 | eqid 2798 | . . 3 ⊢ (Inv‘𝐷) = (Inv‘𝐷) | |
17 | funciso.s | . . . 4 ⊢ 𝐼 = (Iso‘𝐷) | |
18 | 7 | simpld 498 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat) |
19 | funciso.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐼𝑌)) | |
20 | 9, 17, 16, 18, 11, 13, 19 | invisoinvr 17053 | . . 3 ⊢ (𝜑 → 𝑀(𝑋(Inv‘𝐷)𝑌)((𝑋(Inv‘𝐷)𝑌)‘𝑀)) |
21 | 9, 16, 2, 3, 11, 13, 20 | funcinv 17135 | . 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 17028 | 1 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 〈cop 4531 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Catccat 16927 Invcinv 17007 Isociso 17008 Func cfunc 17116 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 df-ixp 8445 df-cat 16931 df-cid 16932 df-sect 17009 df-inv 17010 df-iso 17011 df-func 17120 |
This theorem is referenced by: ffthiso 17191 |
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