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Theorem ciclcl 17064
Description: Isomorphism implies the left side is an object. (Contributed by AV, 5-Apr-2020.)
Assertion
Ref Expression
ciclcl ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑅 ∈ (Base‘𝐶))

Proof of Theorem ciclcl
StepHypRef Expression
1 cicfval 17059 . . . 4 (𝐶 ∈ Cat → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))
21breqd 5041 . . 3 (𝐶 ∈ Cat → (𝑅( ≃𝑐𝐶)𝑆𝑅((Iso‘𝐶) supp ∅)𝑆))
3 isofn 17037 . . . . 5 (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
4 fvex 6658 . . . . . 6 (Base‘𝐶) ∈ V
5 sqxpexg 7457 . . . . . 6 ((Base‘𝐶) ∈ V → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
64, 5mp1i 13 . . . . 5 (𝐶 ∈ Cat → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
7 0ex 5175 . . . . . 6 ∅ ∈ V
87a1i 11 . . . . 5 (𝐶 ∈ Cat → ∅ ∈ V)
9 df-br 5031 . . . . . 6 (𝑅((Iso‘𝐶) supp ∅)𝑆 ↔ ⟨𝑅, 𝑆⟩ ∈ ((Iso‘𝐶) supp ∅))
10 elsuppfn 7821 . . . . . 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⟩) ≠ ∅)))
119, 10syl5bb 286 . . . . 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⟩) ≠ ∅)))
123, 6, 8, 11syl3anc 1368 . . . 4 (𝐶 ∈ Cat → (𝑅((Iso‘𝐶) supp ∅)𝑆 ↔ (⟨𝑅, 𝑆⟩ ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Iso‘𝐶)‘⟨𝑅, 𝑆⟩) ≠ ∅)))
13 opelxp1 5560 . . . . 5 (⟨𝑅, 𝑆⟩ ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑅 ∈ (Base‘𝐶))
1413adantr 484 . . . 4 ((⟨𝑅, 𝑆⟩ ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Iso‘𝐶)‘⟨𝑅, 𝑆⟩) ≠ ∅) → 𝑅 ∈ (Base‘𝐶))
1512, 14syl6bi 256 . . 3 (𝐶 ∈ Cat → (𝑅((Iso‘𝐶) supp ∅)𝑆𝑅 ∈ (Base‘𝐶)))
162, 15sylbid 243 . 2 (𝐶 ∈ Cat → (𝑅( ≃𝑐𝐶)𝑆𝑅 ∈ (Base‘𝐶)))
1716imp 410 1 ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑅 ∈ (Base‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084  wcel 2111  wne 2987  Vcvv 3441  c0 4243  cop 4531   class class class wbr 5030   × cxp 5517   Fn wfn 6319  cfv 6324  (class class class)co 7135   supp csupp 7813  Basecbs 16475  Catccat 16927  Isociso 17008  𝑐 ccic 17057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-supp 7814  df-inv 17010  df-iso 17011  df-cic 17058
This theorem is referenced by:  cicsym  17066  cictr  17067  cicer  17068  initoeu2  17268
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