MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isofval Structured version   Visualization version   GIF version

Theorem isofval 17743
Description: Function value of the function returning the isomorphisms of a category. (Contributed by AV, 5-Apr-2017.)
Assertion
Ref Expression
isofval (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)))
Distinct variable group:   𝑥,𝐶

Proof of Theorem isofval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 df-iso 17735 . 2 Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)))
2 fveq2 6896 . . 3 (𝑐 = 𝐶 → (Inv‘𝑐) = (Inv‘𝐶))
32coeq2d 5865 . 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 6592 . . 3 Fun (𝑥 ∈ V ↦ dom 𝑥)
6 fvexd 6911 . . 3 (𝐶 ∈ Cat → (Inv‘𝐶) ∈ V)
7 cofunexg 7953 . . 3 ((Fun (𝑥 ∈ V ↦ dom 𝑥) ∧ (Inv‘𝐶) ∈ V) → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) ∈ V)
85, 6, 7sylancr 585 . 2 (𝐶 ∈ Cat → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) ∈ V)
91, 3, 4, 8fvmptd3 7027 1 (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3461  cmpt 5232  dom cdm 5678  ccom 5682  Fun wfun 6543  cfv 6549  Catccat 17647  Invcinv 17731  Isociso 17732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-iso 17735
This theorem is referenced by:  isoval  17751  isofn  17761
  Copyright terms: Public domain W3C validator