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Theorem issubc2 16892
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 6237 . . . . 5 (𝐽 Fn (𝑆 × 𝑆) → dom 𝐽 = (𝑆 × 𝑆))
75, 6syl 17 . . . 4 (𝜑 → dom 𝐽 = (𝑆 × 𝑆))
87dmeqd 5573 . . 3 (𝜑 → dom dom 𝐽 = dom (𝑆 × 𝑆))
9 dmxpid 5592 . . 3 dom (𝑆 × 𝑆) = 𝑆
108, 9syl6req 2831 . 2 (𝜑𝑆 = dom dom 𝐽)
111, 2, 3, 4, 10issubc 16891 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 198  wa 386   = wceq 1601  wcel 2107  wral 3090  cop 4404   class class class wbr 4888   × cxp 5355  dom cdm 5357   Fn wfn 6132  cfv 6137  (class class class)co 6924  compcco 16361  Catccat 16721  Idccid 16722  Homf chomf 16723  cat cssc 16863  Subcatcsubc 16865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-pm 8145  df-ixp 8197  df-ssc 16866  df-subc 16868
This theorem is referenced by:  0subcat  16894  catsubcat  16895  subcidcl  16900  subccocl  16901  issubc3  16905  fullsubc  16906  rnghmsubcsetc  43006  rhmsubcsetc  43052  rhmsubcrngc  43058  srhmsubc  43105  rhmsubc  43119  srhmsubcALTV  43123  rhmsubcALTV  43137
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