Theorem homarcl 18123

Description: Reverse closure for an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypothesis

Ref	Expression
homarcl.h	⊢ 𝐻 = (Homa‘𝐶)

Assertion

Ref	Expression
homarcl	⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)

Proof of Theorem homarcl

Dummy variables 𝑥 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0i 4289	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅)
2		homarcl.h	. . . . 5 ⊢ 𝐻 = (Homa‘𝐶)
3		df-homa 18121	. . . . . 6 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
4	3	fvmptndm 7022	. . . . 5 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅)
5	2, 4	eqtrid 2809	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅)
6	5	oveqd 7434	. . 3 ⊢ (¬ 𝐶 ∈ Cat → (𝑋𝐻𝑌) = (𝑋∅𝑌))
7		0ov 7454	. . 3 ⊢ (𝑋∅𝑌) = ∅
8	6, 7	eqtrdi 2813	. 2 ⊢ (¬ 𝐶 ∈ Cat → (𝑋𝐻𝑌) = ∅)
9	1, 8	nsyl2 142	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2145  ∅c0 4282  {csn 4587   ↦ cmpt 5190   × cxp 5657  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  Hom chom 17359  Catccat 17758  Hom_achoma 18118

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-dm 5669  df-iota 6493  df-fv 6545  df-ov 7420  df-homa 18121

This theorem is used by:  homarcl2  18130  homarel  18131  homa1  18132  homahom2  18133  coahom  18165  arwlid  18167  arwrid  18168  arwass  18169