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Theorem cicrcl 17075
Description: Isomorphism implies the right side is an object. (Contributed by AV, 5-Apr-2020.)
Assertion
Ref Expression
cicrcl ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶))

Proof of Theorem cicrcl
StepHypRef Expression
1 cicfval 17069 . . . 4 (𝐶 ∈ Cat → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))
21breqd 5064 . . 3 (𝐶 ∈ Cat → (𝑅( ≃𝑐𝐶)𝑆𝑅((Iso‘𝐶) supp ∅)𝑆))
3 isofn 17047 . . . . 5 (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
4 fvex 6676 . . . . . 6 (Base‘𝐶) ∈ V
5 sqxpexg 7473 . . . . . 6 ((Base‘𝐶) ∈ V → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
64, 5mp1i 13 . . . . 5 (𝐶 ∈ Cat → ((Base‘𝐶) × (Base‘𝐶)) ∈ V)
7 0ex 5198 . . . . . 6 ∅ ∈ V
87a1i 11 . . . . 5 (𝐶 ∈ Cat → ∅ ∈ V)
9 df-br 5054 . . . . . 6 (𝑅((Iso‘𝐶) supp ∅)𝑆 ↔ ⟨𝑅, 𝑆⟩ ∈ ((Iso‘𝐶) supp ∅))
10 elsuppfn 7836 . . . . . 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 opelxp2 5585 . . . . 5 (⟨𝑅, 𝑆⟩ ∈ ((Base‘𝐶) × (Base‘𝐶)) → 𝑆 ∈ (Base‘𝐶))
1413adantr 484 . . . 4 ((⟨𝑅, 𝑆⟩ ∈ ((Base‘𝐶) × (Base‘𝐶)) ∧ ((Iso‘𝐶)‘⟨𝑅, 𝑆⟩) ≠ ∅) → 𝑆 ∈ (Base‘𝐶))
1512, 14syl6bi 256 . . 3 (𝐶 ∈ Cat → (𝑅((Iso‘𝐶) supp ∅)𝑆𝑆 ∈ (Base‘𝐶)))
162, 15sylbid 243 . 2 (𝐶 ∈ Cat → (𝑅( ≃𝑐𝐶)𝑆𝑆 ∈ (Base‘𝐶)))
1716imp 410 1 ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084  wcel 2115  wne 3014  Vcvv 3480  c0 4276  cop 4556   class class class wbr 5053   × cxp 5541   Fn wfn 6340  cfv 6345  (class class class)co 7151   supp csupp 7828  Basecbs 16485  Catccat 16937  Isociso 17018  𝑐 ccic 17067
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7686  df-2nd 7687  df-supp 7829  df-inv 17020  df-iso 17021  df-cic 17068
This theorem is referenced by:  cicsym  17076  cictr  17077  initoeu2  17278
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