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Theorem subcfn 17853
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 2726 . . 3 (Homf𝐶) = (Homf𝐶)
31, 2subcssc 17852 . 2 (𝜑𝐽cat (Homf𝐶))
4 subcfn.2 . 2 (𝜑𝑆 = dom dom 𝐽)
53, 4sscfn1 17826 1 (𝜑𝐽 Fn (𝑆 × 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099   × cxp 5671  dom cdm 5673   Fn wfn 6539  cfv 6544  Homf chomf 17672  Subcatcsubc 17818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7417  df-oprab 7418  df-mpo 7419  df-pm 8848  df-ixp 8917  df-ssc 17819  df-subc 17821
This theorem is referenced by:  subccat  17860  subsubc  17865  funcres  17908  funcres2  17910  idfusubc  17912  subthinc  48395
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