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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 17007 | . 2 ⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐))) | |
2 | fveq2 6663 | . . 3 ⊢ (𝑐 = 𝐶 → (Inv‘𝑐) = (Inv‘𝐶)) | |
3 | 2 | coeq2d 5726 | . 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 6386 | . . 3 ⊢ Fun (𝑥 ∈ V ↦ dom 𝑥) | |
6 | fvexd 6678 | . . 3 ⊢ (𝐶 ∈ Cat → (Inv‘𝐶) ∈ V) | |
7 | cofunexg 7639 | . . 3 ⊢ ((Fun (𝑥 ∈ V ↦ dom 𝑥) ∧ (Inv‘𝐶) ∈ V) → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) ∈ V) | |
8 | 5, 6, 7 | sylancr 587 | . 2 ⊢ (𝐶 ∈ Cat → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) ∈ V) |
9 | 1, 3, 4, 8 | fvmptd3 6783 | 1 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ↦ cmpt 5137 dom cdm 5548 ∘ ccom 5552 Fun wfun 6342 ‘cfv 6348 Catccat 16923 Invcinv 17003 Isociso 17004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-iso 17007 |
This theorem is referenced by: isoval 17023 isofn 17033 |
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