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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 17070 | . . . 4 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅)) | |
2 | 1 | breqd 5080 | . . 3 ⊢ (𝐶 ∈ Cat → (𝑅( ≃𝑐 ‘𝐶)𝑆 ↔ 𝑅((Iso‘𝐶) supp ∅)𝑆)) |
3 | isofn 17048 | . . . . 5 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) | |
4 | fvex 6686 | . . . . . 6 ⊢ (Base‘𝐶) ∈ V | |
5 | sqxpexg 7480 | . . . . . 6 ⊢ ((Base‘𝐶) ∈ V → ((Base‘𝐶) × (Base‘𝐶)) ∈ V) | |
6 | 4, 5 | mp1i 13 | . . . . 5 ⊢ (𝐶 ∈ Cat → ((Base‘𝐶) × (Base‘𝐶)) ∈ V) |
7 | 0ex 5214 | . . . . . 6 ⊢ ∅ ∈ V | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝐶 ∈ Cat → ∅ ∈ V) |
9 | df-br 5070 | . . . . . 6 ⊢ (𝑅((Iso‘𝐶) supp ∅)𝑆 ↔ 〈𝑅, 𝑆〉 ∈ ((Iso‘𝐶) supp ∅)) | |
10 | elsuppfn 7841 | . . . . . 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 285 | . . . . 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 1367 | . . . 4 ⊢ (𝐶 ∈ Cat → (𝑅((Iso‘𝐶) supp ∅)𝑆 ↔ (〈𝑅, 𝑆〉 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Iso‘𝐶)‘〈𝑅, 𝑆〉) ≠ ∅))) |
13 | opelxp1 5599 | . . . . 5 ⊢ (〈𝑅, 𝑆〉 ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑅 ∈ (Base‘𝐶)) | |
14 | 13 | adantr 483 | . . . 4 ⊢ ((〈𝑅, 𝑆〉 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Iso‘𝐶)‘〈𝑅, 𝑆〉) ≠ ∅) → 𝑅 ∈ (Base‘𝐶)) |
15 | 12, 14 | syl6bi 255 | . . 3 ⊢ (𝐶 ∈ Cat → (𝑅((Iso‘𝐶) supp ∅)𝑆 → 𝑅 ∈ (Base‘𝐶))) |
16 | 2, 15 | sylbid 242 | . 2 ⊢ (𝐶 ∈ Cat → (𝑅( ≃𝑐 ‘𝐶)𝑆 → 𝑅 ∈ (Base‘𝐶))) |
17 | 16 | imp 409 | 1 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑅 ∈ (Base‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2113 ≠ wne 3019 Vcvv 3497 ∅c0 4294 〈cop 4576 class class class wbr 5069 × cxp 5556 Fn wfn 6353 ‘cfv 6358 (class class class)co 7159 supp csupp 7833 Basecbs 16486 Catccat 16938 Isociso 17019 ≃𝑐 ccic 17068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-supp 7834 df-inv 17021 df-iso 17022 df-cic 17069 |
This theorem is referenced by: cicsym 17077 cictr 17078 cicer 17079 initoeu2 17279 |
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