Theorem cicfval 16457

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 16456	. . 3 ⊢ ≃𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅))
2	1	a1i 11	. 2 ⊢ (𝐶 ∈ Cat → ≃𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅)))
3		fveq2 6191	. . . 4 ⊢ (𝑐 = 𝐶 → (Iso‘𝑐) = (Iso‘𝐶))
4	3	oveq1d 6665	. . 3 ⊢ (𝑐 = 𝐶 → ((Iso‘𝑐) supp ∅) = ((Iso‘𝐶) supp ∅))
5	4	adantl 482	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑐 = 𝐶) → ((Iso‘𝑐) supp ∅) = ((Iso‘𝐶) supp ∅))
6		id 22	. 2 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
7		ovexd 6680	. 2 ⊢ (𝐶 ∈ Cat → ((Iso‘𝐶) supp ∅) ∈ V)
8	2, 5, 6, 7	fvmptd 6288	1 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650   supp csupp 7295  Catccat 16325  Isociso 16406   ≃_𝑐 ccic 16455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-cic 16456

This theorem is referenced by:  brcic  16458  ciclcl  16462  cicrcl  16463  cicer  16466