Theorem issubc2 17738

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 6581	. . . 4 ⊢ (𝜑 → dom 𝐽 = (𝑆 × 𝑆))
7	6	dmeqd 5840	. . 3 ⊢ (𝜑 → dom dom 𝐽 = dom (𝑆 × 𝑆))
8		dmxpid 5865	. . 3 ⊢ dom (𝑆 × 𝑆) = 𝑆
9	7, 8	eqtr2di 2783	. 2 ⊢ (𝜑 → 𝑆 = dom dom 𝐽)
10	1, 2, 3, 4, 9	issubc 17737	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 1541   ∈ wcel 2111  ∀wral 3047  ⟨cop 4577   class class class wbr 5086   × cxp 5609  dom cdm 5611   Fn wfn 6471  ‘cfv 6476  (class class class)co 7341  compcco 17168  Catccat 17565  Idccid 17566  Hom_f chomf 17567   ⊆_cat cssc 17709  Subcatcsubc 17711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-pm 8748  df-ixp 8817  df-ssc 17712  df-subc 17714

This theorem is referenced by:  0subcat  17740  catsubcat  17741  subcidcl  17746  subccocl  17747  issubc3  17751  fullsubc  17752  rnghmsubcsetc  20543  rhmsubcsetc  20572  rhmsubcrngc  20578  srhmsubc  20590  rhmsubc  20599  rhmsubcALTV  48316  srhmsubcALTV  48356  iinfsubc  49090  discsubc  49096  nelsubc2  49101  imasubc3  49188