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Theorem subcssc 17100
Description: An element in the set of subcategories is a subset of the category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
subcixp.1 (𝜑𝐽 ∈ (Subcat‘𝐶))
subcssc.h 𝐻 = (Homf𝐶)
Assertion
Ref Expression
subcssc (𝜑𝐽cat 𝐻)

Proof of Theorem subcssc
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subcixp.1 . . 3 (𝜑𝐽 ∈ (Subcat‘𝐶))
2 subcssc.h . . . 4 𝐻 = (Homf𝐶)
3 eqid 2826 . . . 4 (Id‘𝐶) = (Id‘𝐶)
4 eqid 2826 . . . 4 (comp‘𝐶) = (comp‘𝐶)
5 subcrcl 17076 . . . . 5 (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
61, 5syl 17 . . . 4 (𝜑𝐶 ∈ Cat)
7 eqidd 2827 . . . 4 (𝜑 → dom dom 𝐽 = dom dom 𝐽)
82, 3, 4, 6, 7issubc 17095 . . 3 (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐽(((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ dom dom 𝐽𝑧 ∈ dom dom 𝐽𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
91, 8mpbid 233 . 2 (𝜑 → (𝐽cat 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐽(((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ dom dom 𝐽𝑧 ∈ dom dom 𝐽𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))))
109simpld 495 1 (𝜑𝐽cat 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wral 3143  cop 4570   class class class wbr 5063  dom cdm 5554  cfv 6352  (class class class)co 7148  compcco 16567  Catccat 16925  Idccid 16926  Homf chomf 16927  cat cssc 17067  Subcatcsubc 17069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-pm 8399  df-ixp 8451  df-ssc 17070  df-subc 17072
This theorem is referenced by:  subcfn  17101  subcss1  17102  subcss2  17103  issubc3  17109  subsubc  17113
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