MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  issubc2 Structured version   Visualization version   GIF version

Theorem issubc2 17100
Description: Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
issubc.h 𝐻 = (Homf𝐶)
issubc.i 1 = (Id‘𝐶)
issubc.o · = (comp‘𝐶)
issubc.c (𝜑𝐶 ∈ Cat)
issubc2.a (𝜑𝐽 Fn (𝑆 × 𝑆))
Assertion
Ref Expression
issubc2 (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
Distinct variable groups:   𝑓,𝑔,𝑥,𝑦,𝑧,𝐶   𝑓,𝐽,𝑔,𝑥,𝑦,𝑧   𝑆,𝑓,𝑔,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑓,𝑔)   · (𝑥,𝑦,𝑧,𝑓,𝑔)   1 (𝑥,𝑦,𝑧,𝑓,𝑔)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑔)

Proof of Theorem issubc2
StepHypRef Expression
1 issubc.h . 2 𝐻 = (Homf𝐶)
2 issubc.i . 2 1 = (Id‘𝐶)
3 issubc.o . 2 · = (comp‘𝐶)
4 issubc.c . 2 (𝜑𝐶 ∈ Cat)
5 issubc2.a . . . . 5 (𝜑𝐽 Fn (𝑆 × 𝑆))
6 fndm 6449 . . . . 5 (𝐽 Fn (𝑆 × 𝑆) → dom 𝐽 = (𝑆 × 𝑆))
75, 6syl 17 . . . 4 (𝜑 → dom 𝐽 = (𝑆 × 𝑆))
87dmeqd 5768 . . 3 (𝜑 → dom dom 𝐽 = dom (𝑆 × 𝑆))
9 dmxpid 5794 . . 3 dom (𝑆 × 𝑆) = 𝑆
108, 9syl6req 2873 . 2 (𝜑𝑆 = dom dom 𝐽)
111, 2, 3, 4, 10issubc 17099 1 (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat 𝐻 ∧ ∀𝑥𝑆 (( 1𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  wral 3138  cop 4566   class class class wbr 5058   × cxp 5547  dom cdm 5549   Fn wfn 6344  cfv 6349  (class class class)co 7150  compcco 16571  Catccat 16929  Idccid 16930  Homf chomf 16931  cat cssc 17071  Subcatcsubc 17073
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 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455
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 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-pm 8403  df-ixp 8456  df-ssc 17074  df-subc 17076
This theorem is referenced by:  0subcat  17102  catsubcat  17103  subcidcl  17108  subccocl  17109  issubc3  17113  fullsubc  17114  rnghmsubcsetc  44242  rhmsubcsetc  44288  rhmsubcrngc  44294  srhmsubc  44341  rhmsubc  44355  srhmsubcALTV  44359  rhmsubcALTV  44373
  Copyright terms: Public domain W3C validator