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 e. ( X H Y ) -> C e. 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 e. ( X H Y ) -> -. ( X H Y ) = (/) )
2		homarcl.h	. . . . 5 |- H = (Homa ` C )
3		df-homa 16676	. . . . . 6 |- Homa = ( c e. Cat |-> ( x e. ( ( Base ` c ) X. ( Base ` c ) ) |-> ( { x } X. ( ( Hom ` c ) ` x ) ) ) )
4	3	fvmptndm 6308	. . . . 5 |- ( -. C e. Cat -> (Homa ` C ) = (/) )
5	2, 4	syl5eq 2668	. . . 4 |- ( -. C e. Cat -> H = (/) )
6	5	oveqd 6667	. . 3 |- ( -. C e. Cat -> ( X H Y ) = ( X (/) Y ) )
7		0ov 6682	. . 3 |- ( X (/) Y ) = (/)
8	6, 7	syl6eq 2672	. 2 |- ( -. C e. Cat -> ( X H Y ) = (/) )
9	1, 8	nsyl2 142	1 |- ( F e. ( X H Y ) -> C e. Cat )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   (/)c0 3915   {csn 4177   |-> cmpt 4729   X. 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