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Theorem cicfval 16853
 Description: The set of isomorphic objects of the category 𝑐. (Contributed by AV, 4-Apr-2020.)
Assertion
Ref Expression
cicfval (𝐶 ∈ Cat → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))

Proof of Theorem cicfval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 df-cic 16852 . 2 𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅))
2 fveq2 6448 . . . 4 (𝑐 = 𝐶 → (Iso‘𝑐) = (Iso‘𝐶))
32oveq1d 6939 . . 3 (𝑐 = 𝐶 → ((Iso‘𝑐) supp ∅) = ((Iso‘𝐶) supp ∅))
43adantl 475 . 2 ((𝐶 ∈ Cat ∧ 𝑐 = 𝐶) → ((Iso‘𝑐) supp ∅) = ((Iso‘𝐶) supp ∅))
5 id 22 . 2 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
6 ovexd 6958 . 2 (𝐶 ∈ Cat → ((Iso‘𝐶) supp ∅) ∈ V)
71, 4, 5, 6fvmptd2 6551 1 (𝐶 ∈ Cat → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2107  Vcvv 3398  ∅c0 4141  ‘cfv 6137  (class class class)co 6924   supp csupp 7578  Catccat 16721  Isociso 16802   ≃𝑐 ccic 16851 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-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140 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-ral 3095  df-rex 3096  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-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  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-iota 6101  df-fun 6139  df-fv 6145  df-ov 6927  df-cic 16852 This theorem is referenced by:  brcic  16854  ciclcl  16858  cicrcl  16859  cicer  16862
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