Theorem homarcl 17743

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 4267	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅)
2		homarcl.h	. . . . 5 ⊢ 𝐻 = (Homa‘𝐶)
3		df-homa 17741	. . . . . 6 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
4	3	fvmptndm 6905	. . . . 5 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅)
5	2, 4	eqtrid 2790	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅)
6	5	oveqd 7292	. . 3 ⊢ (¬ 𝐶 ∈ Cat → (𝑋𝐻𝑌) = (𝑋∅𝑌))
7		0ov 7312	. . 3 ⊢ (𝑋∅𝑌) = ∅
8	6, 7	eqtrdi 2794	. 2 ⊢ (¬ 𝐶 ∈ Cat → (𝑋𝐻𝑌) = ∅)
9	1, 8	nsyl2 141	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ∈ wcel 2106  ∅c0 4256  {csn 4561   ↦ cmpt 5157   × cxp 5587  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  Hom chom 16973  Catccat 17373  Hom_achoma 17738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-dm 5599  df-iota 6391  df-fv 6441  df-ov 7278  df-homa 17741

This theorem is referenced by:  homarcl2  17750  homarel  17751  homa1  17752  homahom2  17753  coahom  17785  arwlid  17787  arwrid  17788  arwass  17789