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Theorem subcfn 16267
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 2606 . . 3 (Homf𝐶) = (Homf𝐶)
31, 2subcssc 16266 . 2 (𝜑𝐽cat (Homf𝐶))
4 subcfn.2 . 2 (𝜑𝑆 = dom dom 𝐽)
53, 4sscfn1 16243 1 (𝜑𝐽 Fn (𝑆 × 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976   × cxp 5023  dom cdm 5025   Fn wfn 5782  cfv 5787  Homf chomf 16093  Subcatcsubc 16235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-pm 7721  df-ixp 7769  df-ssc 16236  df-subc 16238
This theorem is referenced by:  subccat  16274  subsubc  16279  funcres  16322  funcres2  16324  idfusubc  41655
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