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Theorem subcidcl 16327
 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 subcidcl.x . 2 (𝜑𝑋𝑆)
2 subcidcl.j . . . . 5 (𝜑𝐽 ∈ (Subcat‘𝐶))
3 eqid 2610 . . . . . 6 (Homf𝐶) = (Homf𝐶)
4 subcidcl.1 . . . . . 6 1 = (Id‘𝐶)
5 eqid 2610 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
6 subcrcl 16299 . . . . . . 7 (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
72, 6syl 17 . . . . . 6 (𝜑𝐶 ∈ Cat)
8 subcidcl.2 . . . . . 6 (𝜑𝐽 Fn (𝑆 × 𝑆))
93, 4, 5, 7, 8issubc2 16319 . . . . 5 (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat (Homf𝐶) ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
102, 9mpbid 221 . . . 4 (𝜑 → (𝐽cat (Homf𝐶) ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))))
1110simprd 478 . . 3 (𝜑 → ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
12 simpl 472 . . . 4 ((( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)) → ( 1𝑥) ∈ (𝑥𝐽𝑥))
1312ralimi 2936 . . 3 (∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)) → ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥))
1411, 13syl 17 . 2 (𝜑 → ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥))
15 fveq2 6103 . . . 4 (𝑥 = 𝑋 → ( 1𝑥) = ( 1𝑋))
16 id 22 . . . . 5 (𝑥 = 𝑋𝑥 = 𝑋)
1716, 16oveq12d 6567 . . . 4 (𝑥 = 𝑋 → (𝑥𝐽𝑥) = (𝑋𝐽𝑋))
1815, 17eleq12d 2682 . . 3 (𝑥 = 𝑋 → (( 1𝑥) ∈ (𝑥𝐽𝑥) ↔ ( 1𝑋) ∈ (𝑋𝐽𝑋)))
1918rspcv 3278 . 2 (𝑋𝑆 → (∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) → ( 1𝑋) ∈ (𝑋𝐽𝑋)))
201, 14, 19sylc 63 1 (𝜑 → ( 1𝑋) ∈ (𝑋𝐽𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⟨cop 4131   class class class wbr 4583   × cxp 5036   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  compcco 15780  Catccat 16148  Idccid 16149  Homf chomf 16150   ⊆cat cssc 16290  Subcatcsubc 16292 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-pm 7747  df-ixp 7795  df-ssc 16293  df-subc 16295 This theorem is referenced by:  subccatid  16329  issubc3  16332  funcres  16379
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