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

Theorem isofval 16806
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 16798 . 2 Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)))
2 fveq2 6448 . . 3 (𝑐 = 𝐶 → (Inv‘𝑐) = (Inv‘𝐶))
32coeq2d 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)
85, 6, 7sylancr 581 . 2 (𝐶 ∈ Cat → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) ∈ V)
91, 3, 4, 8fvmptd3 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
  Copyright terms: Public domain W3C validator