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Mirrors > Home > MPE Home > Th. List > cicfval | Structured version Visualization version GIF version |
Description: The set of isomorphic objects of the category 𝑐. (Contributed by AV, 4-Apr-2020.) |
Ref | Expression |
---|---|
cicfval | ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cic 16852 | . 2 ⊢ ≃𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅)) | |
2 | fveq2 6448 | . . . 4 ⊢ (𝑐 = 𝐶 → (Iso‘𝑐) = (Iso‘𝐶)) | |
3 | 2 | oveq1d 6939 | . . 3 ⊢ (𝑐 = 𝐶 → ((Iso‘𝑐) supp ∅) = ((Iso‘𝐶) supp ∅)) |
4 | 3 | adantl 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) | |
7 | 1, 4, 5, 6 | fvmptd2 6551 | 1 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((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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