Theorem issubc2 17805

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 6626	. . . 4 ⊢ (𝜑 → dom 𝐽 = (𝑆 × 𝑆))
7	6	dmeqd 5872	. . 3 ⊢ (𝜑 → dom dom 𝐽 = dom (𝑆 × 𝑆))
8		dmxpid 5897	. . 3 ⊢ dom (𝑆 × 𝑆) = 𝑆
9	7, 8	eqtr2di 2782	. 2 ⊢ (𝜑 → 𝑆 = dom dom 𝐽)
10	1, 2, 3, 4, 9	issubc 17804	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ⟨cop 4598   class class class wbr 5110   × cxp 5639  dom cdm 5641   Fn wfn 6509  ‘cfv 6514  (class class class)co 7390  compcco 17239  Catccat 17632  Idccid 17633  Hom_f chomf 17634   ⊆_cat cssc 17776  Subcatcsubc 17778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-pm 8805  df-ixp 8874  df-ssc 17779  df-subc 17781

This theorem is referenced by:  0subcat  17807  catsubcat  17808  subcidcl  17813  subccocl  17814  issubc3  17818  fullsubc  17819  rnghmsubcsetc  20549  rhmsubcsetc  20578  rhmsubcrngc  20584  srhmsubc  20596  rhmsubc  20605  rhmsubcALTV  48277  srhmsubcALTV  48317  iinfsubc  49051  discsubc  49057  nelsubc2  49062  imasubc3  49149