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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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