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Mirrors > Home > MPE Home > Th. List > isofval | Structured version Visualization version GIF version |
Description: Function value of the function returning the isomorphisms of a category. (Contributed by AV, 5-Apr-2017.) |
Ref | Expression |
---|---|
isofval | ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iso 17011 | . 2 ⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐))) | |
2 | fveq2 6645 | . . 3 ⊢ (𝑐 = 𝐶 → (Inv‘𝑐) = (Inv‘𝐶)) | |
3 | 2 | coeq2d 5697 | . 2 ⊢ (𝑐 = 𝐶 → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶))) |
4 | id 22 | . 2 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat) | |
5 | funmpt 6362 | . . 3 ⊢ Fun (𝑥 ∈ V ↦ dom 𝑥) | |
6 | fvexd 6660 | . . 3 ⊢ (𝐶 ∈ Cat → (Inv‘𝐶) ∈ V) | |
7 | cofunexg 7632 | . . 3 ⊢ ((Fun (𝑥 ∈ V ↦ dom 𝑥) ∧ (Inv‘𝐶) ∈ V) → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) ∈ V) | |
8 | 5, 6, 7 | sylancr 590 | . 2 ⊢ (𝐶 ∈ Cat → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) ∈ V) |
9 | 1, 3, 4, 8 | fvmptd3 6768 | 1 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ↦ cmpt 5110 dom cdm 5519 ∘ ccom 5523 Fun wfun 6318 ‘cfv 6324 Catccat 16927 Invcinv 17007 Isociso 17008 |
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-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-iso 17011 |
This theorem is referenced by: isoval 17027 isofn 17037 |
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