Theorem isnat2 16608

Description: Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
natfval.1	|- N = ( C Nat D )
natfval.b	|- B = ( Base ` C )
natfval.h	|- H = ( Hom ` C )
natfval.j	|- J = ( Hom ` D )
natfval.o	|- .x. = (comp ` D )
isnat2.f	|- ( ph -> F e. ( C Func D ) )
isnat2.g	|- ( ph -> G e. ( C Func D ) )

Assertion

Ref	Expression
isnat2	|- ( ph -> ( A e. ( F N G ) <-> ( A e. X_ x e. B ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) /\ A. x e. B A. y e. B A. h e. ( x H y ) ( ( A ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. .x. ( ( 1st ` G ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` h ) ) = ( ( ( x ( 2nd ` G ) y ) ` h ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` G ) ` y ) ) ( A ` x ) ) ) ) )

Distinct variable groups:   x, h, y, A   x, B, y   C, h, x, y   h, F, x, y   h, G, x, y   h, H   ph, h, x, y   D, h, x, y

Allowed substitution hints:   B( h)   .x. ( x, y, h)   H( x, y)   J( x, y, h)   N( x, y, h)

Proof of Theorem isnat2

Step	Hyp	Ref	Expression
1		relfunc 16522	. . . . 5 |- Rel ( C Func D )
2		isnat2.f	. . . . 5 |- ( ph -> F e. ( C Func D ) )
3		1st2nd 7214	. . . . 5 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
4	1, 2, 3	sylancr 695	. . . 4 |- ( ph -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
5		isnat2.g	. . . . 5 |- ( ph -> G e. ( C Func D ) )
6		1st2nd 7214	. . . . 5 |- ( ( Rel ( C Func D ) /\ G e. ( C Func D ) ) -> G = <. ( 1st ` G ) , ( 2nd ` G ) >. )
7	1, 5, 6	sylancr 695	. . . 4 |- ( ph -> G = <. ( 1st ` G ) , ( 2nd ` G ) >. )
8	4, 7	oveq12d 6668	. . 3 |- ( ph -> ( F N G ) = ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )
9	8	eleq2d 2687	. 2 |- ( ph -> ( A e. ( F N G ) <-> A e. ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) ) )
10		natfval.1	. . 3 |- N = ( C Nat D )
11		natfval.b	. . 3 |- B = ( Base ` C )
12		natfval.h	. . 3 |- H = ( Hom ` C )
13		natfval.j	. . 3 |- J = ( Hom ` D )
14		natfval.o	. . 3 |- .x. = (comp ` D )
15		1st2ndbr 7217	. . . 4 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
16	1, 2, 15	sylancr 695	. . 3 |- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
17		1st2ndbr 7217	. . . 4 |- ( ( Rel ( C Func D ) /\ G e. ( C Func D ) ) -> ( 1st ` G ) ( C Func D ) ( 2nd ` G ) )
18	1, 5, 17	sylancr 695	. . 3 |- ( ph -> ( 1st ` G ) ( C Func D ) ( 2nd ` G ) )
19	10, 11, 12, 13, 14, 16, 18	isnat 16607	. 2 |- ( ph -> ( A e. ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) <-> ( A e. X_ x e. B ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) /\ A. x e. B A. y e. B A. h e. ( x H y ) ( ( A ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. .x. ( ( 1st ` G ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` h ) ) = ( ( ( x ( 2nd ` G ) y ) ` h ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` G ) ` y ) ) ( A ` x ) ) ) ) )
20	9, 19	bitrd 268	1 |- ( ph -> ( A e. ( F N G ) <-> ( A e. X_ x e. B ( ( ( 1st ` F ) ` x ) J ( ( 1st ` G ) ` x ) ) /\ A. x e. B A. y e. B A. h e. ( x H y ) ( ( A ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. .x. ( ( 1st ` G ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` h ) ) = ( ( ( x ( 2nd ` G ) y ) ` h ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. .x. ( ( 1st ` G ) ` y ) ) ( A ` x ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   <.cop 4183   class class class wbr 4653   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   X_cixp 7908   Basecbs 15857   Hom chom 15952  compcco 15953   Func cfunc 16514   Nat cnat 16601

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-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-ixp 7909  df-func 16518  df-nat 16603

This theorem is referenced by:  fuccocl  16624  fucidcl  16625  invfuc  16634  curf2cl  16871  yonedalem4c  16917  yonedalem3  16920