Theorem homarcl 16678

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 3920	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅)
2		homarcl.h	. . . . 5 ⊢ 𝐻 = (Homa‘𝐶)
3		df-homa 16676	. . . . . 6 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
4	3	fvmptndm 6308	. . . . 5 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅)
5	2, 4	syl5eq 2668	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅)
6	5	oveqd 6667	. . 3 ⊢ (¬ 𝐶 ∈ Cat → (𝑋𝐻𝑌) = (𝑋∅𝑌))
7		0ov 6682	. . 3 ⊢ (𝑋∅𝑌) = ∅
8	6, 7	syl6eq 2672	. 2 ⊢ (¬ 𝐶 ∈ Cat → (𝑋𝐻𝑌) = ∅)
9	1, 8	nsyl2 142	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1483   ∈ wcel 1990  ∅c0 3915  {csn 4177   ↦ cmpt 4729   × cxp 5112  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  Hom chom 15952  Catccat 16325  Hom_achoma 16673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653  df-homa 16676

This theorem is referenced by:  homarcl2  16685  homarel  16686  homa1  16687  homahom2  16688  coahom  16720  arwlid  16722  arwrid  16723  arwass  16724