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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 16457 | . . . 4 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅)) | |
2 | 1 | breqd 4664 | . . 3 ⊢ (𝐶 ∈ Cat → (𝑅( ≃𝑐 ‘𝐶)𝑆 ↔ 𝑅((Iso‘𝐶) supp ∅)𝑆)) |
3 | isofn 16435 | . . . . 5 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) | |
4 | fvex 6201 | . . . . . 6 ⊢ (Base‘𝐶) ∈ V | |
5 | sqxpexg 6963 | . . . . . 6 ⊢ ((Base‘𝐶) ∈ V → ((Base‘𝐶) × (Base‘𝐶)) ∈ V) | |
6 | 4, 5 | mp1i 13 | . . . . 5 ⊢ (𝐶 ∈ Cat → ((Base‘𝐶) × (Base‘𝐶)) ∈ V) |
7 | 0ex 4790 | . . . . . 6 ⊢ ∅ ∈ V | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝐶 ∈ Cat → ∅ ∈ V) |
9 | df-br 4654 | . . . . . 6 ⊢ (𝑅((Iso‘𝐶) supp ∅)𝑆 ↔ 〈𝑅, 𝑆〉 ∈ ((Iso‘𝐶) supp ∅)) | |
10 | elsuppfn 7303 | . . . . . 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 272 | . . . . 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 1326 | . . . 4 ⊢ (𝐶 ∈ Cat → (𝑅((Iso‘𝐶) supp ∅)𝑆 ↔ (〈𝑅, 𝑆〉 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Iso‘𝐶)‘〈𝑅, 𝑆〉) ≠ ∅))) |
13 | opelxp2 5151 | . . . . 5 ⊢ (〈𝑅, 𝑆〉 ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑆 ∈ (Base‘𝐶)) | |
14 | 13 | adantr 481 | . . . 4 ⊢ ((〈𝑅, 𝑆〉 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Iso‘𝐶)‘〈𝑅, 𝑆〉) ≠ ∅) → 𝑆 ∈ (Base‘𝐶)) |
15 | 12, 14 | syl6bi 243 | . . 3 ⊢ (𝐶 ∈ Cat → (𝑅((Iso‘𝐶) supp ∅)𝑆 → 𝑆 ∈ (Base‘𝐶))) |
16 | 2, 15 | sylbid 230 | . 2 ⊢ (𝐶 ∈ Cat → (𝑅( ≃𝑐 ‘𝐶)𝑆 → 𝑆 ∈ (Base‘𝐶))) |
17 | 16 | imp 445 | 1 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∅c0 3915 〈cop 4183 class class class wbr 4653 × cxp 5112 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 supp csupp 7295 Basecbs 15857 Catccat 16325 Isociso 16406 ≃𝑐 ccic 16455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-supp 7296 df-inv 16408 df-iso 16409 df-cic 16456 |
This theorem is referenced by: cicsym 16464 cictr 16465 initoeu2 16666 |
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