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Mirrors > Home > MPE Home > Th. List > ciclcl | Structured version Visualization version GIF version |
Description: Isomorphism implies the left side is an object. (Contributed by AV, 5-Apr-2020.) |
Ref | Expression |
---|---|
ciclcl | ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑅 ∈ (Base‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cicfval 17256 | . . . 4 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅)) | |
2 | 1 | breqd 5050 | . . 3 ⊢ (𝐶 ∈ Cat → (𝑅( ≃𝑐 ‘𝐶)𝑆 ↔ 𝑅((Iso‘𝐶) supp ∅)𝑆)) |
3 | isofn 17234 | . . . . 5 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) | |
4 | fvexd 6710 | . . . . 5 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) ∈ V) | |
5 | 0ex 5185 | . . . . . 6 ⊢ ∅ ∈ V | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (𝐶 ∈ Cat → ∅ ∈ V) |
7 | df-br 5040 | . . . . . 6 ⊢ (𝑅((Iso‘𝐶) supp ∅)𝑆 ↔ 〈𝑅, 𝑆〉 ∈ ((Iso‘𝐶) supp ∅)) | |
8 | elsuppfng 7890 | . . . . . 6 ⊢ (((Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ (Iso‘𝐶) ∈ V ∧ ∅ ∈ V) → (〈𝑅, 𝑆〉 ∈ ((Iso‘𝐶) supp ∅) ↔ (〈𝑅, 𝑆〉 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Iso‘𝐶)‘〈𝑅, 𝑆〉) ≠ ∅))) | |
9 | 7, 8 | syl5bb 286 | . . . . 5 ⊢ (((Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ (Iso‘𝐶) ∈ V ∧ ∅ ∈ V) → (𝑅((Iso‘𝐶) supp ∅)𝑆 ↔ (〈𝑅, 𝑆〉 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Iso‘𝐶)‘〈𝑅, 𝑆〉) ≠ ∅))) |
10 | 3, 4, 6, 9 | syl3anc 1373 | . . . 4 ⊢ (𝐶 ∈ Cat → (𝑅((Iso‘𝐶) supp ∅)𝑆 ↔ (〈𝑅, 𝑆〉 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Iso‘𝐶)‘〈𝑅, 𝑆〉) ≠ ∅))) |
11 | opelxp1 5577 | . . . . 5 ⊢ (〈𝑅, 𝑆〉 ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑅 ∈ (Base‘𝐶)) | |
12 | 11 | adantr 484 | . . . 4 ⊢ ((〈𝑅, 𝑆〉 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Iso‘𝐶)‘〈𝑅, 𝑆〉) ≠ ∅) → 𝑅 ∈ (Base‘𝐶)) |
13 | 10, 12 | syl6bi 256 | . . 3 ⊢ (𝐶 ∈ Cat → (𝑅((Iso‘𝐶) supp ∅)𝑆 → 𝑅 ∈ (Base‘𝐶))) |
14 | 2, 13 | sylbid 243 | . 2 ⊢ (𝐶 ∈ Cat → (𝑅( ≃𝑐 ‘𝐶)𝑆 → 𝑅 ∈ (Base‘𝐶))) |
15 | 14 | imp 410 | 1 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑅 ∈ (Base‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2112 ≠ wne 2932 Vcvv 3398 ∅c0 4223 〈cop 4533 class class class wbr 5039 × cxp 5534 Fn wfn 6353 ‘cfv 6358 (class class class)co 7191 supp csupp 7881 Basecbs 16666 Catccat 17121 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-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-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-supp 7882 df-inv 17207 df-iso 17208 df-cic 17255 |
This theorem is referenced by: cicsym 17263 cictr 17264 cicer 17265 initoeu2 17476 |
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