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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 17058 | . 2 ⊢ ≃𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅)) | |
2 | fveq2 6645 | . . 3 ⊢ (𝑐 = 𝐶 → (Iso‘𝑐) = (Iso‘𝐶)) | |
3 | 2 | oveq1d 7150 | . 2 ⊢ (𝑐 = 𝐶 → ((Iso‘𝑐) supp ∅) = ((Iso‘𝐶) supp ∅)) |
4 | id 22 | . 2 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat) | |
5 | ovexd 7170 | . 2 ⊢ (𝐶 ∈ Cat → ((Iso‘𝐶) supp ∅) ∈ V) | |
6 | 1, 3, 4, 5 | fvmptd3 6768 | 1 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 ‘cfv 6324 (class class class)co 7135 supp csupp 7813 Catccat 16927 Isociso 17008 ≃𝑐 ccic 17057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-cic 17058 |
This theorem is referenced by: brcic 17060 ciclcl 17064 cicrcl 17065 cicer 17068 |
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