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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 17794 | . 2 ⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐))) | |
| 2 | fveq2 6871 | . . 3 ⊢ (𝑐 = 𝐶 → (Inv‘𝑐) = (Inv‘𝐶)) | |
| 3 | 2 | coeq2d 5838 | . 2 ⊢ (𝑐 = 𝐶 → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶))) |
| 4 | id 23 | . 2 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat) | |
| 5 | funmpt 6563 | . . 3 ⊢ Fun (𝑥 ∈ V ↦ dom 𝑥) | |
| 6 | fvexd 6886 | . . 3 ⊢ (𝐶 ∈ Cat → (Inv‘𝐶) ∈ V) | |
| 7 | cofunexg 7934 | . . 3 ⊢ ((Fun (𝑥 ∈ V ↦ dom 𝑥) ∧ (Inv‘𝐶) ∈ V) → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) ∈ V) | |
| 8 | 5, 6, 7 | sylancr 598 | . 2 ⊢ (𝐶 ∈ Cat → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) ∈ V) |
| 9 | 1, 3, 4, 8 | fvmptd3 7003 | 1 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1563 ∈ wcel 2145 Vcvv 3457 ↦ cmpt 5185 dom cdm 5651 ∘ ccom 5655 Fun wfun 6519 ‘cfv 6525 Catccat 17708 Invcinv 17790 Isociso 17791 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5231 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3080 df-rex 3090 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5105 df-opab 5167 df-mpt 5186 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-iso 17794 |
| This theorem is referenced by: isoval 17810 isofn 17820 isofnALT 49661 |
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