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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 17782 | . 2 ⊢ ≃𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅)) | |
2 | fveq2 6896 | . . 3 ⊢ (𝑐 = 𝐶 → (Iso‘𝑐) = (Iso‘𝐶)) | |
3 | 2 | oveq1d 7434 | . 2 ⊢ (𝑐 = 𝐶 → ((Iso‘𝑐) supp ∅) = ((Iso‘𝐶) supp ∅)) |
4 | id 22 | . 2 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat) | |
5 | ovexd 7454 | . 2 ⊢ (𝐶 ∈ Cat → ((Iso‘𝐶) supp ∅) ∈ V) | |
6 | 1, 3, 4, 5 | fvmptd3 7027 | 1 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ∅c0 4322 ‘cfv 6549 (class class class)co 7419 supp csupp 8165 Catccat 17647 Isociso 17732 ≃𝑐 ccic 17781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-ov 7422 df-cic 17782 |
This theorem is referenced by: brcic 17784 ciclcl 17788 cicrcl 17789 cicer 17792 |
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