Theorem cicfval 18022

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 18021	. 2 ⊢ ≃𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅))
2		fveq2 6916	. . 3 ⊢ (𝑐 = 𝐶 → (Iso‘𝑐) = (Iso‘𝐶))
3	2	oveq1d 7466	. 2 ⊢ (𝑐 = 𝐶 → ((Iso‘𝑐) supp ∅) = ((Iso‘𝐶) supp ∅))
4		id 23	. 2 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
5		ovexd 7486	. 2 ⊢ (𝐶 ∈ Cat → ((Iso‘𝐶) supp ∅) ∈ V)
6	1, 3, 4, 5	fvmptd3 7048	1 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ∅c0 4281  ‘cfv 6566  (class class class)co 7451   supp csupp 8214  Catccat 17888  Isociso 17971   ≃_𝑐 ccic 18020

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-cic 18021

This theorem is used by:  brcic  18023  ciclcl  18027  cicrcl  18028  cicer  18031  relcic  50490