Theorem issubc2 17467

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	5	fndmd 6522	. . . 4 ⊢ (𝜑 → dom 𝐽 = (𝑆 × 𝑆))
7	6	dmeqd 5803	. . 3 ⊢ (𝜑 → dom dom 𝐽 = dom (𝑆 × 𝑆))
8		dmxpid 5828	. . 3 ⊢ dom (𝑆 × 𝑆) = 𝑆
9	7, 8	eqtr2di 2796	. 2 ⊢ (𝜑 → 𝑆 = dom dom 𝐽)
10	1, 2, 3, 4, 9	issubc 17466	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 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ⟨cop 4564   class class class wbr 5070   × cxp 5578  dom cdm 5580   Fn wfn 6413  ‘cfv 6418  (class class class)co 7255  compcco 16900  Catccat 17290  Idccid 17291  Hom_f chomf 17292   ⊆_cat cssc 17436  Subcatcsubc 17438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-pm 8576  df-ixp 8644  df-ssc 17439  df-subc 17441

This theorem is referenced by:  0subcat  17469  catsubcat  17470  subcidcl  17475  subccocl  17476  issubc3  17480  fullsubc  17481  rnghmsubcsetc  45423  rhmsubcsetc  45469  rhmsubcrngc  45475  srhmsubc  45522  rhmsubc  45536  srhmsubcALTV  45540  rhmsubcALTV  45554