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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 16279 | . . 3 ⊢ ≃𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅)) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐶 ∈ Cat → ≃𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅))) |
3 | fveq2 6103 | . . . 4 ⊢ (𝑐 = 𝐶 → (Iso‘𝑐) = (Iso‘𝐶)) | |
4 | 3 | oveq1d 6564 | . . 3 ⊢ (𝑐 = 𝐶 → ((Iso‘𝑐) supp ∅) = ((Iso‘𝐶) supp ∅)) |
5 | 4 | adantl 481 | . 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑐 = 𝐶) → ((Iso‘𝑐) supp ∅) = ((Iso‘𝐶) supp ∅)) |
6 | id 22 | . 2 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat) | |
7 | ovex 6577 | . . 3 ⊢ ((Iso‘𝐶) supp ∅) ∈ V | |
8 | 7 | a1i 11 | . 2 ⊢ (𝐶 ∈ Cat → ((Iso‘𝐶) supp ∅) ∈ V) |
9 | 2, 5, 6, 8 | fvmptd 6197 | 1 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 supp csupp 7182 Catccat 16148 Isociso 16229 ≃𝑐 ccic 16278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-cic 16279 |
This theorem is referenced by: brcic 16281 ciclcl 16285 cicrcl 16286 cicer 16289 |
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