Theorem issubc2 16543

Description: Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
issubc.h	⊢ 𝐻 = (Homf ‘𝐶)
issubc.i	⊢ 1 = (Id‘𝐶)
issubc.o	⊢ · = (comp‘𝐶)
issubc.c	⊢ (𝜑 → 𝐶 ∈ Cat)
issubc2.a	⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))

Assertion

Ref	Expression
issubc2	⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (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

Step	Hyp	Ref	Expression
1		issubc.h	. 2 ⊢ 𝐻 = (Homf ‘𝐶)
2		issubc.i	. 2 ⊢ 1 = (Id‘𝐶)
3		issubc.o	. 2 ⊢ · = (comp‘𝐶)
4		issubc.c	. 2 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		issubc2.a	. . . . 5 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))
6		fndm 6028	. . . . 5 ⊢ (𝐽 Fn (𝑆 × 𝑆) → dom 𝐽 = (𝑆 × 𝑆))
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → dom 𝐽 = (𝑆 × 𝑆))
8	7	dmeqd 5358	. . 3 ⊢ (𝜑 → dom dom 𝐽 = dom (𝑆 × 𝑆))
9		dmxpid 5377	. . 3 ⊢ dom (𝑆 × 𝑆) = 𝑆
10	8, 9	syl6req 2702	. 2 ⊢ (𝜑 → 𝑆 = dom dom 𝐽)
11	1, 2, 3, 4, 10	issubc 16542	1 ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat 𝐻 ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ⟨cop 4216   class class class wbr 4685   × cxp 5141  dom cdm 5143   Fn wfn 5921  ‘cfv 5926  (class class class)co 6690  compcco 16000  Catccat 16372  Idccid 16373  Hom_f chomf 16374   ⊆_cat cssc 16514  Subcatcsubc 16516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-pm 7902  df-ixp 7951  df-ssc 16517  df-subc 16519

This theorem is referenced by:  0subcat  16545  catsubcat  16546  subcidcl  16551  subccocl  16552  issubc3  16556  fullsubc  16557  rnghmsubcsetc  42302  rhmsubcsetc  42348  rhmsubcrngc  42354  srhmsubc  42401  rhmsubc  42415  srhmsubcALTV  42419  rhmsubcALTV  42433