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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 16798 | . 2 ⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐))) | |
2 | fveq2 6448 | . . 3 ⊢ (𝑐 = 𝐶 → (Inv‘𝑐) = (Inv‘𝐶)) | |
3 | 2 | coeq2d 5532 | . 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 6175 | . . 3 ⊢ Fun (𝑥 ∈ V ↦ dom 𝑥) | |
6 | fvexd 6463 | . . 3 ⊢ (𝐶 ∈ Cat → (Inv‘𝐶) ∈ V) | |
7 | cofunexg 7411 | . . 3 ⊢ ((Fun (𝑥 ∈ V ↦ dom 𝑥) ∧ (Inv‘𝐶) ∈ V) → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) ∈ V) | |
8 | 5, 6, 7 | sylancr 581 | . 2 ⊢ (𝐶 ∈ Cat → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) ∈ V) |
9 | 1, 3, 4, 8 | fvmptd3 6566 | 1 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ↦ cmpt 4967 dom cdm 5357 ∘ ccom 5361 Fun wfun 6131 ‘cfv 6137 Catccat 16714 Invcinv 16794 Isociso 16795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-iso 16798 |
This theorem is referenced by: isoval 16814 isofn 16824 |
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