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Theorem isofval 17015
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 17007 . 2 Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)))
2 fveq2 6663 . . 3 (𝑐 = 𝐶 → (Inv‘𝑐) = (Inv‘𝐶))
32coeq2d 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)
85, 6, 7sylancr 587 . 2 (𝐶 ∈ Cat → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) ∈ V)
91, 3, 4, 8fvmptd3 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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