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Theorem subcfn 17111
Description: An element in the set of subcategories is a binary function. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
subcixp.1 (𝜑𝐽 ∈ (Subcat‘𝐶))
subcfn.2 (𝜑𝑆 = dom dom 𝐽)
Assertion
Ref Expression
subcfn (𝜑𝐽 Fn (𝑆 × 𝑆))

Proof of Theorem subcfn
StepHypRef Expression
1 subcixp.1 . . 3 (𝜑𝐽 ∈ (Subcat‘𝐶))
2 eqid 2824 . . 3 (Homf𝐶) = (Homf𝐶)
31, 2subcssc 17110 . 2 (𝜑𝐽cat (Homf𝐶))
4 subcfn.2 . 2 (𝜑𝑆 = dom dom 𝐽)
53, 4sscfn1 17087 1 (𝜑𝐽 Fn (𝑆 × 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115   × cxp 5540  dom cdm 5542   Fn wfn 6338  cfv 6343  Homf chomf 16937  Subcatcsubc 17079
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 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  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 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-pm 8405  df-ixp 8458  df-ssc 17080  df-subc 17082
This theorem is referenced by:  subccat  17118  subsubc  17123  funcres  17166  funcres2  17168  idfusubc  44421
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