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Theorem subcidcl 16711
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 6333 . . 3 (𝑥 = 𝑋 → ( 1𝑥) = ( 1𝑋))
2 id 22 . . . 4 (𝑥 = 𝑋𝑥 = 𝑋)
32, 2oveq12d 6814 . . 3 (𝑥 = 𝑋 → (𝑥𝐽𝑥) = (𝑋𝐽𝑋))
41, 3eleq12d 2844 . 2 (𝑥 = 𝑋 → (( 1𝑥) ∈ (𝑥𝐽𝑥) ↔ ( 1𝑋) ∈ (𝑋𝐽𝑋)))
5 subcidcl.j . . . . 5 (𝜑𝐽 ∈ (Subcat‘𝐶))
6 eqid 2771 . . . . . 6 (Homf𝐶) = (Homf𝐶)
7 subcidcl.1 . . . . . 6 1 = (Id‘𝐶)
8 eqid 2771 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
9 subcrcl 16683 . . . . . . 7 (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
105, 9syl 17 . . . . . 6 (𝜑𝐶 ∈ Cat)
11 subcidcl.2 . . . . . 6 (𝜑𝐽 Fn (𝑆 × 𝑆))
126, 7, 8, 10, 11issubc2 16703 . . . . 5 (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat (Homf𝐶) ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
135, 12mpbid 222 . . . 4 (𝜑 → (𝐽cat (Homf𝐶) ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))))
1413simprd 483 . . 3 (𝜑 → ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
15 simpl 468 . . . 4 ((( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)) → ( 1𝑥) ∈ (𝑥𝐽𝑥))
1615ralimi 3101 . . 3 (∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)) → ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥))
1714, 16syl 17 . 2 (𝜑 → ∀𝑥𝑆 ( 1𝑥) ∈ (𝑥𝐽𝑥))
18 subcidcl.x . 2 (𝜑𝑋𝑆)
194, 17, 18rspcdva 3466 1 (𝜑 → ( 1𝑋) ∈ (𝑋𝐽𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  cop 4323   class class class wbr 4787   × cxp 5248   Fn wfn 6025  cfv 6030  (class class class)co 6796  compcco 16161  Catccat 16532  Idccid 16533  Homf chomf 16534  cat cssc 16674  Subcatcsubc 16676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-pm 8016  df-ixp 8067  df-ssc 16677  df-subc 16679
This theorem is referenced by:  subccatid  16713  issubc3  16716  funcres  16763
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