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Theorem subcidcl 16273
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 2609 . . . . . 6 (Homf𝐶) = (Homf𝐶)
4 subcidcl.1 . . . . . 6 1 = (Id‘𝐶)
5 eqid 2609 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
6 subcrcl 16245 . . . . . . 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 16265 . . . . 5 (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat (Homf𝐶) ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
102, 9mpbid 220 . . . 4 (𝜑 → (𝐽cat (Homf𝐶) ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))))
1110simprd 477 . . 3 (𝜑 → ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
12 simpl 471 . . . 4 ((( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)) → ( 1𝑥) ∈ (𝑥𝐽𝑥))
1312ralimi 2935 . . 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 6088 . . . 4 (𝑥 = 𝑋 → ( 1𝑥) = ( 1𝑋))
16 id 22 . . . . 5 (𝑥 = 𝑋𝑥 = 𝑋)
1716, 16oveq12d 6545 . . . 4 (𝑥 = 𝑋 → (𝑥𝐽𝑥) = (𝑋𝐽𝑋))
1815, 17eleq12d 2681 . . 3 (𝑥 = 𝑋 → (( 1𝑥) ∈ (𝑥𝐽𝑥) ↔ ( 1𝑋) ∈ (𝑋𝐽𝑋)))
1918rspcv 3277 . 2 (𝑋𝑆 → (∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥) → ( 1𝑋) ∈ (𝑋𝐽𝑋)))
201, 14, 19sylc 62 1 (𝜑 → ( 1𝑋) ∈ (𝑋𝐽𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2895  cop 4130   class class class wbr 4577   × cxp 5026   Fn wfn 5785  cfv 5790  (class class class)co 6527  compcco 15726  Catccat 16094  Idccid 16095  Homf chomf 16096  cat cssc 16236  Subcatcsubc 16238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-pm 7724  df-ixp 7772  df-ssc 16239  df-subc 16241
This theorem is referenced by:  subccatid  16275  issubc3  16278  funcres  16325
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