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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 17021 | . 2 ⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐))) | |
2 | fveq2 6672 | . . 3 ⊢ (𝑐 = 𝐶 → (Inv‘𝑐) = (Inv‘𝐶)) | |
3 | 2 | coeq2d 5735 | . 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 6395 | . . 3 ⊢ Fun (𝑥 ∈ V ↦ dom 𝑥) | |
6 | fvexd 6687 | . . 3 ⊢ (𝐶 ∈ Cat → (Inv‘𝐶) ∈ V) | |
7 | cofunexg 7652 | . . 3 ⊢ ((Fun (𝑥 ∈ V ↦ dom 𝑥) ∧ (Inv‘𝐶) ∈ V) → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) ∈ V) | |
8 | 5, 6, 7 | sylancr 589 | . 2 ⊢ (𝐶 ∈ Cat → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) ∈ V) |
9 | 1, 3, 4, 8 | fvmptd3 6793 | 1 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ↦ cmpt 5148 dom cdm 5557 ∘ ccom 5561 Fun wfun 6351 ‘cfv 6357 Catccat 16937 Invcinv 17017 Isociso 17018 |
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-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-iso 17021 |
This theorem is referenced by: isoval 17037 isofn 17047 |
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