![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cicrcl | Structured version Visualization version GIF version |
Description: Isomorphism implies the right side is an object. (Contributed by AV, 5-Apr-2020.) |
Ref | Expression |
---|---|
cicrcl | ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cicfval 16846 | . . . 4 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅)) | |
2 | 1 | breqd 4899 | . . 3 ⊢ (𝐶 ∈ Cat → (𝑅( ≃𝑐 ‘𝐶)𝑆 ↔ 𝑅((Iso‘𝐶) supp ∅)𝑆)) |
3 | isofn 16824 | . . . . 5 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) | |
4 | fvex 6461 | . . . . . 6 ⊢ (Base‘𝐶) ∈ V | |
5 | sqxpexg 7243 | . . . . . 6 ⊢ ((Base‘𝐶) ∈ V → ((Base‘𝐶) × (Base‘𝐶)) ∈ V) | |
6 | 4, 5 | mp1i 13 | . . . . 5 ⊢ (𝐶 ∈ Cat → ((Base‘𝐶) × (Base‘𝐶)) ∈ V) |
7 | 0ex 5028 | . . . . . 6 ⊢ ∅ ∈ V | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝐶 ∈ Cat → ∅ ∈ V) |
9 | df-br 4889 | . . . . . 6 ⊢ (𝑅((Iso‘𝐶) supp ∅)𝑆 ↔ 〈𝑅, 𝑆〉 ∈ ((Iso‘𝐶) supp ∅)) | |
10 | elsuppfn 7586 | . . . . . 6 ⊢ (((Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Base‘𝐶) × (Base‘𝐶)) ∈ V ∧ ∅ ∈ V) → (〈𝑅, 𝑆〉 ∈ ((Iso‘𝐶) supp ∅) ↔ (〈𝑅, 𝑆〉 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Iso‘𝐶)‘〈𝑅, 𝑆〉) ≠ ∅))) | |
11 | 9, 10 | syl5bb 275 | . . . . 5 ⊢ (((Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Base‘𝐶) × (Base‘𝐶)) ∈ V ∧ ∅ ∈ V) → (𝑅((Iso‘𝐶) supp ∅)𝑆 ↔ (〈𝑅, 𝑆〉 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Iso‘𝐶)‘〈𝑅, 𝑆〉) ≠ ∅))) |
12 | 3, 6, 8, 11 | syl3anc 1439 | . . . 4 ⊢ (𝐶 ∈ Cat → (𝑅((Iso‘𝐶) supp ∅)𝑆 ↔ (〈𝑅, 𝑆〉 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Iso‘𝐶)‘〈𝑅, 𝑆〉) ≠ ∅))) |
13 | opelxp2 5399 | . . . . 5 ⊢ (〈𝑅, 𝑆〉 ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑆 ∈ (Base‘𝐶)) | |
14 | 13 | adantr 474 | . . . 4 ⊢ ((〈𝑅, 𝑆〉 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Iso‘𝐶)‘〈𝑅, 𝑆〉) ≠ ∅) → 𝑆 ∈ (Base‘𝐶)) |
15 | 12, 14 | syl6bi 245 | . . 3 ⊢ (𝐶 ∈ Cat → (𝑅((Iso‘𝐶) supp ∅)𝑆 → 𝑆 ∈ (Base‘𝐶))) |
16 | 2, 15 | sylbid 232 | . 2 ⊢ (𝐶 ∈ Cat → (𝑅( ≃𝑐 ‘𝐶)𝑆 → 𝑆 ∈ (Base‘𝐶))) |
17 | 16 | imp 397 | 1 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 ∈ wcel 2107 ≠ wne 2969 Vcvv 3398 ∅c0 4141 〈cop 4404 class class class wbr 4888 × cxp 5355 Fn wfn 6132 ‘cfv 6137 (class class class)co 6924 supp csupp 7578 Basecbs 16259 Catccat 16714 Isociso 16795 ≃𝑐 ccic 16844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-1st 7447 df-2nd 7448 df-supp 7579 df-inv 16797 df-iso 16798 df-cic 16845 |
This theorem is referenced by: cicsym 16853 cictr 16854 initoeu2 17055 |
Copyright terms: Public domain | W3C validator |