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Theorem subcidcl 17113
 Description: The identity of the original category is contained in each subcategory. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
subcidcl.j (𝜑𝐽 ∈ (Subcat‘𝐶))
subcidcl.2 (𝜑𝐽 Fn (𝑆 × 𝑆))
subcidcl.x (𝜑𝑋𝑆)
subcidcl.1 1 = (Id‘𝐶)
Assertion
Ref Expression
subcidcl (𝜑 → ( 1𝑋) ∈ (𝑋𝐽𝑋))

Proof of Theorem subcidcl
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6669 . . 3 (𝑥 = 𝑋 → ( 1𝑥) = ( 1𝑋))
2 id 22 . . . 4 (𝑥 = 𝑋𝑥 = 𝑋)
32, 2oveq12d 7173 . . 3 (𝑥 = 𝑋 → (𝑥𝐽𝑥) = (𝑋𝐽𝑋))
41, 3eleq12d 2907 . 2 (𝑥 = 𝑋 → (( 1𝑥) ∈ (𝑥𝐽𝑥) ↔ ( 1𝑋) ∈ (𝑋𝐽𝑋)))
5 subcidcl.j . . . 4 (𝜑𝐽 ∈ (Subcat‘𝐶))
6 eqid 2821 . . . . 5 (Homf𝐶) = (Homf𝐶)
7 subcidcl.1 . . . . 5 1 = (Id‘𝐶)
8 eqid 2821 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
9 subcrcl 17085 . . . . . 6 (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
105, 9syl 17 . . . . 5 (𝜑𝐶 ∈ Cat)
11 subcidcl.2 . . . . 5 (𝜑𝐽 Fn (𝑆 × 𝑆))
126, 7, 8, 10, 11issubc2 17105 . . . 4 (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat (Homf𝐶) ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
135, 12mpbid 234 . . 3 (𝜑 → (𝐽cat (Homf𝐶) ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))))
14 simpl 485 . . . 4 ((( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)) → ( 1𝑥) ∈ (𝑥𝐽𝑥))
1514ralimi 3160 . . 3 (∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)) → ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥))
1613, 15simpl2im 506 . 2 (𝜑 → ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥))
17 subcidcl.x . 2 (𝜑𝑋𝑆)
184, 16, 17rspcdva 3624 1 (𝜑 → ( 1𝑋) ∈ (𝑋𝐽𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ⟨cop 4572   class class class wbr 5065   × cxp 5552   Fn wfn 6349  ‘cfv 6354  (class class class)co 7155  compcco 16576  Catccat 16934  Idccid 16935  Homf chomf 16936   ⊆cat cssc 17076  Subcatcsubc 17078 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  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 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-pm 8408  df-ixp 8461  df-ssc 17079  df-subc 17081 This theorem is referenced by:  subccatid  17115  issubc3  17118  funcres  17165
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