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Mirrors > Home > MPE Home > Th. List > cicref | Structured version Visualization version GIF version |
Description: Isomorphism is reflexive. (Contributed by AV, 5-Apr-2020.) |
Ref | Expression |
---|---|
cicref | ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝑂( ≃𝑐 ‘𝐶)𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . 2 ⊢ (Iso‘𝐶) = (Iso‘𝐶) | |
2 | eqid 2736 | . 2 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | simpl 486 | . 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat) | |
4 | simpr 488 | . 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝑂 ∈ (Base‘𝐶)) | |
5 | eqid 2736 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
6 | 2, 5, 3, 4 | idiso 17247 | . 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑂) ∈ (𝑂(Iso‘𝐶)𝑂)) |
7 | 1, 2, 3, 4, 4, 6 | brcici 17259 | 1 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (Base‘𝐶)) → 𝑂( ≃𝑐 ‘𝐶)𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2112 class class class wbr 5039 ‘cfv 6358 Basecbs 16666 Catccat 17121 Idccid 17122 Isociso 17205 ≃𝑐 ccic 17254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-supp 7882 df-cat 17125 df-cid 17126 df-sect 17206 df-inv 17207 df-iso 17208 df-cic 17255 |
This theorem is referenced by: cicer 17265 |
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