Theorem cicfval 17845

Description: The set of isomorphic objects of the category 𝑐. (Contributed by AV, 4-Apr-2020.)

Assertion

Ref	Expression
cicfval	⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅))

Proof of Theorem cicfval

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cic 17844	. 2 ⊢ ≃𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅))
2		fveq2 6907	. . 3 ⊢ (𝑐 = 𝐶 → (Iso‘𝑐) = (Iso‘𝐶))
3	2	oveq1d 7446	. 2 ⊢ (𝑐 = 𝐶 → ((Iso‘𝑐) supp ∅) = ((Iso‘𝐶) supp ∅))
4		id 22	. 2 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
5		ovexd 7466	. 2 ⊢ (𝐶 ∈ Cat → ((Iso‘𝐶) supp ∅) ∈ V)
6	1, 3, 4, 5	fvmptd3 7039	1 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ∅c0 4339  ‘cfv 6563  (class class class)co 7431   supp csupp 8184  Catccat 17709  Isociso 17794   ≃_𝑐 ccic 17843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-cic 17844

This theorem is referenced by:  brcic  17846  ciclcl  17850  cicrcl  17851  cicer  17854