Theorem cicfval 17061

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 17060	. 2 ⊢ ≃𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅))
2		fveq2 6664	. . 3 ⊢ (𝑐 = 𝐶 → (Iso‘𝑐) = (Iso‘𝐶))
3	2	oveq1d 7165	. 2 ⊢ (𝑐 = 𝐶 → ((Iso‘𝑐) supp ∅) = ((Iso‘𝐶) supp ∅))
4		id 22	. 2 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
5		ovexd 7185	. 2 ⊢ (𝐶 ∈ Cat → ((Iso‘𝐶) supp ∅) ∈ V)
6	1, 3, 4, 5	fvmptd3 6785	1 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  Vcvv 3494  ∅c0 4290  ‘cfv 6349  (class class class)co 7150   supp csupp 7824  Catccat 16929  Isociso 17010   ≃_𝑐 ccic 17059

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357  df-ov 7153  df-cic 17060

This theorem is referenced by:  brcic  17062  ciclcl  17066  cicrcl  17067  cicer  17070