Theorem fullfunc 16566

Description: A full functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)

Assertion

Ref	Expression
fullfunc	|- ( C Full D ) C_ ( C Func D )

Proof of Theorem fullfunc

Dummy variables c d f g x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . 4 |- ( c = C -> ( c Full d ) = ( C Full d ) )
2		oveq1 6657	. . . 4 |- ( c = C -> ( c Func d ) = ( C Func d ) )
3	1, 2	sseq12d 3634	. . 3 |- ( c = C -> ( ( c Full d ) C_ ( c Func d ) <-> ( C Full d ) C_ ( C Func d ) ) )
4		oveq2 6658	. . . 4 |- ( d = D -> ( C Full d ) = ( C Full D ) )
5		oveq2 6658	. . . 4 |- ( d = D -> ( C Func d ) = ( C Func D ) )
6	4, 5	sseq12d 3634	. . 3 |- ( d = D -> ( ( C Full d ) C_ ( C Func d ) <-> ( C Full D ) C_ ( C Func D ) ) )
7		ovex 6678	. . . . . 6 |- ( c Func d ) e. _V
8		simpl 473	. . . . . . . 8 |- ( ( f ( c Func d ) g /\ A. x e. ( Base ` c ) A. y e. ( Base ` c ) ran ( x g y ) = ( ( f ` x ) ( Hom ` d ) ( f ` y ) ) ) -> f ( c Func d ) g )
9	8	ssopab2i 5003	. . . . . . 7 |- { <. f , g >. | ( f ( c Func d ) g /\ A. x e. ( Base ` c ) A. y e. ( Base ` c ) ran ( x g y ) = ( ( f ` x ) ( Hom ` d ) ( f ` y ) ) ) } C_ { <. f , g >. | f ( c Func d ) g }
10		opabss 4714	. . . . . . 7 |- { <. f , g >. | f ( c Func d ) g } C_ ( c Func d )
11	9, 10	sstri 3612	. . . . . 6 |- { <. f , g >. | ( f ( c Func d ) g /\ A. x e. ( Base ` c ) A. y e. ( Base ` c ) ran ( x g y ) = ( ( f ` x ) ( Hom ` d ) ( f ` y ) ) ) } C_ ( c Func d )
12	7, 11	ssexi 4803	. . . . 5 |- { <. f , g >. | ( f ( c Func d ) g /\ A. x e. ( Base ` c ) A. y e. ( Base ` c ) ran ( x g y ) = ( ( f ` x ) ( Hom ` d ) ( f ` y ) ) ) } e. _V
13		df-full 16564	. . . . . 6 |- Full = ( c e. Cat , d e. Cat |-> { <. f , g >. | ( f ( c Func d ) g /\ A. x e. ( Base ` c ) A. y e. ( Base ` c ) ran ( x g y ) = ( ( f ` x ) ( Hom ` d ) ( f ` y ) ) ) } )
14	13	ovmpt4g 6783	. . . . 5 |- ( ( c e. Cat /\ d e. Cat /\ { <. f , g >. | ( f ( c Func d ) g /\ A. x e. ( Base ` c ) A. y e. ( Base ` c ) ran ( x g y ) = ( ( f ` x ) ( Hom ` d ) ( f ` y ) ) ) } e. _V ) -> ( c Full d ) = { <. f , g >. | ( f ( c Func d ) g /\ A. x e. ( Base ` c ) A. y e. ( Base ` c ) ran ( x g y ) = ( ( f ` x ) ( Hom ` d ) ( f ` y ) ) ) } )
15	12, 14	mp3an3 1413	. . . 4 |- ( ( c e. Cat /\ d e. Cat ) -> ( c Full d ) = { <. f , g >. | ( f ( c Func d ) g /\ A. x e. ( Base ` c ) A. y e. ( Base ` c ) ran ( x g y ) = ( ( f ` x ) ( Hom ` d ) ( f ` y ) ) ) } )
16	15, 11	syl6eqss 3655	. . 3 |- ( ( c e. Cat /\ d e. Cat ) -> ( c Full d ) C_ ( c Func d ) )
17	3, 6, 16	vtocl2ga 3274	. 2 |- ( ( C e. Cat /\ D e. Cat ) -> ( C Full D ) C_ ( C Func D ) )
18	13	mpt2ndm0 6875	. . 3 |- ( -. ( C e. Cat /\ D e. Cat ) -> ( C Full D ) = (/) )
19		0ss 3972	. . 3 |- (/) C_ ( C Func D )
20	18, 19	syl6eqss 3655	. 2 |- ( -. ( C e. Cat /\ D e. Cat ) -> ( C Full D ) C_ ( C Func D ) )
21	17, 20	pm2.61i 176	1 |- ( C Full D ) C_ ( C Func D )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653   {copab 4712   ran crn 5115   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952   Catccat 16325   Func cfunc 16514   Full cful 16562

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-sbc 3436  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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-full 16564

This theorem is referenced by:  relfull  16568  isfull  16570  fulloppc  16582  cofull  16594  catcisolem  16756  catciso  16757