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Theorem issubc2 16319
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 5904 . . . . 5 (𝐽 Fn (𝑆 × 𝑆) → dom 𝐽 = (𝑆 × 𝑆))
75, 6syl 17 . . . 4 (𝜑 → dom 𝐽 = (𝑆 × 𝑆))
87dmeqd 5248 . . 3 (𝜑 → dom dom 𝐽 = dom (𝑆 × 𝑆))
9 dmxpid 5266 . . 3 dom (𝑆 × 𝑆) = 𝑆
108, 9syl6req 2661 . 2 (𝜑𝑆 = dom dom 𝐽)
111, 2, 3, 4, 10issubc 16318 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 195  wa 383   = wceq 1475  wcel 1977  wral 2896  cop 4131   class class class wbr 4583   × cxp 5036  dom cdm 5038   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:  0subcat  16321  catsubcat  16322  subcidcl  16327  subccocl  16328  issubc3  16332  fullsubc  16333  rnghmsubcsetc  41769  rhmsubcsetc  41815  rhmsubcrngc  41821  srhmsubc  41868  rhmsubc  41882  srhmsubcALTV  41887  rhmsubcALTV  41901
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