Theorem homarcl 17993

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 4275	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅)
2		homarcl.h	. . . . 5 ⊢ 𝐻 = (Homa‘𝐶)
3		df-homa 17991	. . . . . 6 ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥))))
4	3	fvmptndm 6974	. . . . 5 ⊢ (¬ 𝐶 ∈ Cat → (Homa‘𝐶) = ∅)
5	2, 4	eqtrid 2787	. . . 4 ⊢ (¬ 𝐶 ∈ Cat → 𝐻 = ∅)
6	5	oveqd 7380	. . 3 ⊢ (¬ 𝐶 ∈ Cat → (𝑋𝐻𝑌) = (𝑋∅𝑌))
7		0ov 7400	. . 3 ⊢ (𝑋∅𝑌) = ∅
8	6, 7	eqtrdi 2791	. 2 ⊢ (¬ 𝐶 ∈ Cat → (𝑋𝐻𝑌) = ∅)
9	1, 8	nsyl2 141	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1547   ∈ wcel 2119  ∅c0 4268  {csn 4562   ↦ cmpt 5160   × cxp 5623  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  Hom chom 17229  Catccat 17628  Hom_achoma 17988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-dm 5635  df-iota 6448  df-fv 6500  df-ov 7366  df-homa 17991

This theorem is referenced by:  homarcl2  18000  homarel  18001  homa1  18002  homahom2  18003  coahom  18035  arwlid  18037  arwrid  18038  arwass  18039