Theorem homarcl 17990

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 4303	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅)
2		homarcl.h	. . . . 5 ⊢ 𝐻 = (Homa‘𝐶)
3		df-homa 17988	. . . . . 6 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
4	3	fvmptndm 6999	. . . . 5 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅)
5	2, 4	eqtrid 2776	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅)
6	5	oveqd 7404	. . 3 ⊢ (¬ 𝐶 ∈ Cat → (𝑋𝐻𝑌) = (𝑋∅𝑌))
7		0ov 7424	. . 3 ⊢ (𝑋∅𝑌) = ∅
8	6, 7	eqtrdi 2780	. 2 ⊢ (¬ 𝐶 ∈ Cat → (𝑋𝐻𝑌) = ∅)
9	1, 8	nsyl2 141	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109  ∅c0 4296  {csn 4589   ↦ cmpt 5188   × cxp 5636  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  Hom chom 17231  Catccat 17625  Hom_achoma 17985

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-dm 5648  df-iota 6464  df-fv 6519  df-ov 7390  df-homa 17988

This theorem is referenced by:  homarcl2  17997  homarel  17998  homa1  17999  homahom2  18000  coahom  18032  arwlid  18034  arwrid  18035  arwass  18036