Theorem homaf 16680

Description: Functionality of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
homarcl.h	|- H = (Homa ` C )
homafval.b	|- B = ( Base ` C )
homafval.c	|- ( ph -> C e. Cat )

Assertion

Ref	Expression
homaf	|- ( ph -> H : ( B X. B ) --> ~P ( ( B X. B ) X. _V ) )

Proof of Theorem homaf

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		snssi 4339	. . . . . 6 |- ( x e. ( B X. B ) -> { x } C_ ( B X. B ) )
2	1	adantl 482	. . . . 5 |- ( ( ph /\ x e. ( B X. B ) ) -> { x } C_ ( B X. B ) )
3		ssv 3625	. . . . 5 |- ( ( Hom ` C ) ` x ) C_ _V
4		xpss12 5225	. . . . 5 |- ( ( { x } C_ ( B X. B ) /\ ( ( Hom ` C ) ` x ) C_ _V ) -> ( { x } X. ( ( Hom ` C ) ` x ) ) C_ ( ( B X. B ) X. _V ) )
5	2, 3, 4	sylancl 694	. . . 4 |- ( ( ph /\ x e. ( B X. B ) ) -> ( { x } X. ( ( Hom ` C ) ` x ) ) C_ ( ( B X. B ) X. _V ) )
6		snex 4908	. . . . . 6 |- { x } e. _V
7		fvex 6201	. . . . . 6 |- ( ( Hom ` C ) ` x ) e. _V
8	6, 7	xpex 6962	. . . . 5 |- ( { x } X. ( ( Hom ` C ) ` x ) ) e. _V
9	8	elpw 4164	. . . 4 |- ( ( { x } X. ( ( Hom ` C ) ` x ) ) e. ~P ( ( B X. B ) X. _V ) <-> ( { x } X. ( ( Hom ` C ) ` x ) ) C_ ( ( B X. B ) X. _V ) )
10	5, 9	sylibr 224	. . 3 |- ( ( ph /\ x e. ( B X. B ) ) -> ( { x } X. ( ( Hom ` C ) ` x ) ) e. ~P ( ( B X. B ) X. _V ) )
11		eqid 2622	. . 3 |- ( x e. ( B X. B ) |-> ( { x } X. ( ( Hom ` C ) ` x ) ) ) = ( x e. ( B X. B ) |-> ( { x } X. ( ( Hom ` C ) ` x ) ) )
12	10, 11	fmptd 6385	. 2 |- ( ph -> ( x e. ( B X. B ) |-> ( { x } X. ( ( Hom ` C ) ` x ) ) ) : ( B X. B ) --> ~P ( ( B X. B ) X. _V ) )
13		homarcl.h	. . . 4 |- H = (Homa ` C )
14		homafval.b	. . . 4 |- B = ( Base ` C )
15		homafval.c	. . . 4 |- ( ph -> C e. Cat )
16		eqid 2622	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
17	13, 14, 15, 16	homafval 16679	. . 3 |- ( ph -> H = ( x e. ( B X. B ) |-> ( { x } X. ( ( Hom ` C ) ` x ) ) ) )
18	17	feq1d 6030	. 2 |- ( ph -> ( H : ( B X. B ) --> ~P ( ( B X. B ) X. _V ) <-> ( x e. ( B X. B ) |-> ( { x } X. ( ( Hom ` C ) ` x ) ) ) : ( B X. B ) --> ~P ( ( B X. B ) X. _V ) ) )
19	12, 18	mpbird 247	1 |- ( ph -> H : ( B X. B ) --> ~P ( ( B X. B ) X. _V ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   {csn 4177   |-> cmpt 4729   X. cxp 5112   -->wf 5884   ` cfv 5888   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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-homa 16676

This theorem is referenced by:  homarcl2  16685  homarel  16686  arwhoma  16695