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Theorem cicfval 17065
Description: The set of isomorphic objects of the category 𝑐. (Contributed by AV, 4-Apr-2020.)
Assertion
Ref Expression
cicfval (𝐶 ∈ Cat → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))

Proof of Theorem cicfval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 df-cic 17064 . 2 𝑐 = (𝑐 ∈ Cat ↦ ((Iso‘𝑐) supp ∅))
2 fveq2 6659 . . 3 (𝑐 = 𝐶 → (Iso‘𝑐) = (Iso‘𝐶))
32oveq1d 7161 . 2 (𝑐 = 𝐶 → ((Iso‘𝑐) supp ∅) = ((Iso‘𝐶) supp ∅))
4 id 22 . 2 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
5 ovexd 7181 . 2 (𝐶 ∈ Cat → ((Iso‘𝐶) supp ∅) ∈ V)
61, 3, 4, 5fvmptd3 6780 1 (𝐶 ∈ Cat → ( ≃𝑐𝐶) = ((Iso‘𝐶) supp ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3480  c0 4276  cfv 6344  (class class class)co 7146   supp csupp 7822  Catccat 16933  Isociso 17014  𝑐 ccic 17063
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-sep 5190  ax-nul 5197  ax-pr 5318
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-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  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-iota 6303  df-fun 6346  df-fv 6352  df-ov 7149  df-cic 17064
This theorem is referenced by:  brcic  17066  ciclcl  17070  cicrcl  17071  cicer  17074
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