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Theorem isofval 16464
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 16456 . . 3 Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)))
21a1i 11 . 2 (𝐶 ∈ Cat → Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐))))
3 fveq2 6229 . . . 4 (𝑐 = 𝐶 → (Inv‘𝑐) = (Inv‘𝐶))
43coeq2d 5317 . . 3 (𝑐 = 𝐶 → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)))
54adantl 481 . 2 ((𝐶 ∈ Cat ∧ 𝑐 = 𝐶) → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)))
6 id 22 . 2 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
7 funmpt 5964 . . 3 Fun (𝑥 ∈ V ↦ dom 𝑥)
8 fvexd 6241 . . 3 (𝐶 ∈ Cat → (Inv‘𝐶) ∈ V)
9 cofunexg 7172 . . 3 ((Fun (𝑥 ∈ V ↦ dom 𝑥) ∧ (Inv‘𝐶) ∈ V) → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) ∈ V)
107, 8, 9sylancr 696 . 2 (𝐶 ∈ Cat → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) ∈ V)
112, 5, 6, 10fvmptd 6327 1 (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  cmpt 4762  dom cdm 5143  ccom 5147  Fun wfun 5920  cfv 5926  Catccat 16372  Invcinv 16452  Isociso 16453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-iso 16456
This theorem is referenced by:  isoval  16472  isofn  16482
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