Theorem cicrcl 17072

Description: Isomorphism implies the right side is an object. (Contributed by AV, 5-Apr-2020.)

Assertion

Ref	Expression
cicrcl	⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶))

Proof of Theorem cicrcl

Step	Hyp	Ref	Expression
1		cicfval 17066	. . . 4 ⊢ (𝐶 ∈ Cat → ( ≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅))
2	1	breqd 5076	. . 3 ⊢ (𝐶 ∈ Cat → (𝑅( ≃𝑐 ‘𝐶)𝑆 ↔ 𝑅((Iso‘𝐶) supp ∅)𝑆))
3		isofn 17044	. . . . 5 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
4		fvex 6682	. . . . . 6 ⊢ (Base‘𝐶) ∈ V
5		sqxpexg 7476	. . . . . 6 ⊢ ((Base‘𝐶) ∈ V → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
6	4, 5	mp1i 13	. . . . 5 ⊢ (𝐶 ∈ Cat → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
7		0ex 5210	. . . . . 6 ⊢ ∅ ∈ V
8	7	a1i 11	. . . . 5 ⊢ (𝐶 ∈ Cat → ∅ ∈ V)
9		df-br 5066	. . . . . 6 ⊢ (𝑅((Iso‘𝐶) supp ∅)𝑆 ↔ ⟨𝑅, 𝑆⟩ ∈ ((Iso‘𝐶) supp ∅))
10		elsuppfn 7837	. . . . . 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		opelxp2 5596	. . . . 5 ⊢ (⟨𝑅, 𝑆⟩ ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑆 ∈ (Base‘𝐶))
14	13	adantr 483	. . . 4 ⊢ ((⟨𝑅, 𝑆⟩ ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Iso‘𝐶)‘⟨𝑅, 𝑆⟩) ≠ ∅) → 𝑆 ∈ (Base‘𝐶))
15	12, 14	syl6bi 255	. . 3 ⊢ (𝐶 ∈ Cat → (𝑅((Iso‘𝐶) supp ∅)𝑆 → 𝑆 ∈ (Base‘𝐶)))
16	2, 15	sylbid 242	. 2 ⊢ (𝐶 ∈ Cat → (𝑅( ≃𝑐 ‘𝐶)𝑆 → 𝑆 ∈ (Base‘𝐶)))
17	16	imp 409	1 ⊢ ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐 ‘𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   ∈ wcel 2110   ≠ wne 3016  Vcvv 3494  ∅c0 4290  ⟨cop 4572   class class class wbr 5065   × cxp 5552   Fn wfn 6349  ‘cfv 6354  (class class class)co 7155   supp csupp 7829  Basecbs 16482  Catccat 16934  Isociso 17015   ≃_𝑐 ccic 17064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-supp 7830  df-inv 17017  df-iso 17018  df-cic 17065

This theorem is referenced by:  cicsym  17073  cictr  17074  initoeu2  17275