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Theorem isofval 16411
Description: Function value of the function returning the isomorphisms of a category. (Contributed by AV, 5-Apr-2017.)
Assertion
Ref Expression
isofval (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)))
Distinct variable group:   𝑥,𝐶

Proof of Theorem isofval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 df-iso 16403 . . 3 Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)))
21a1i 11 . 2 (𝐶 ∈ Cat → Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐))))
3 fveq2 6189 . . . 4 (𝑐 = 𝐶 → (Inv‘𝑐) = (Inv‘𝐶))
43coeq2d 5282 . . 3 (𝑐 = 𝐶 → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)))
54adantl 482 . 2 ((𝐶 ∈ Cat ∧ 𝑐 = 𝐶) → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)))
6 id 22 . 2 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
7 funmpt 5924 . . 3 Fun (𝑥 ∈ V ↦ dom 𝑥)
8 fvexd 6201 . . 3 (𝐶 ∈ Cat → (Inv‘𝐶) ∈ V)
9 cofunexg 7127 . . 3 ((Fun (𝑥 ∈ V ↦ dom 𝑥) ∧ (Inv‘𝐶) ∈ V) → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) ∈ V)
107, 8, 9sylancr 695 . 2 (𝐶 ∈ Cat → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) ∈ V)
112, 5, 6, 10fvmptd 6286 1 (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989  Vcvv 3198  cmpt 4727  dom cdm 5112  ccom 5116  Fun wfun 5880  cfv 5886  Catccat 16319  Invcinv 16399  Isociso 16400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-iso 16403
This theorem is referenced by:  isoval  16419  isofn  16429
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