Theorem issubc2 17794

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 6597	. . . 4 ⊢ (𝜑 → dom 𝐽 = (𝑆 × 𝑆))
7	6	dmeqd 5854	. . 3 ⊢ (𝜑 → dom dom 𝐽 = dom (𝑆 × 𝑆))
8		dmxpid 5879	. . 3 ⊢ dom (𝑆 × 𝑆) = 𝑆
9	7, 8	eqtr2di 2789	. 2 ⊢ (𝜑 → 𝑆 = dom dom 𝐽)
10	1, 2, 3, 4, 9	issubc 17793	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 1542   ∈ wcel 2114  ∀wral 3052  ⟨cop 4574   class class class wbr 5086   × cxp 5622  dom cdm 5624   Fn wfn 6487  ‘cfv 6492  (class class class)co 7360  compcco 17223  Catccat 17621  Idccid 17622  Hom_f chomf 17623   ⊆_cat cssc 17765  Subcatcsubc 17767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-pm 8769  df-ixp 8839  df-ssc 17768  df-subc 17770

This theorem is referenced by:  0subcat  17796  catsubcat  17797  subcidcl  17802  subccocl  17803  issubc3  17807  fullsubc  17808  rnghmsubcsetc  20601  rhmsubcsetc  20630  rhmsubcrngc  20636  srhmsubc  20648  rhmsubc  20657  rhmsubcALTV  48773  srhmsubcALTV  48813  iinfsubc  49545  discsubc  49551  nelsubc2  49556  imasubc3  49643