Theorem natffn 16609

Description: The natural transformation set operation is a well-defined function. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypothesis

Ref	Expression
natrcl.1	|- N = ( C Nat D )

Assertion

Ref	Expression
natffn	|- N Fn ( ( C Func D ) X. ( C Func D ) )

Proof of Theorem natffn

Dummy variables x f y a g h r s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		natrcl.1	. . 3 |- N = ( C Nat D )
2		eqid 2622	. . 3 |- ( Base ` C ) = ( Base ` C )
3		eqid 2622	. . 3 |- ( Hom ` C ) = ( Hom ` C )
4		eqid 2622	. . 3 |- ( Hom ` D ) = ( Hom ` D )
5		eqid 2622	. . 3 |- (comp ` D ) = (comp ` D )
6	1, 2, 3, 4, 5	natfval 16606	. 2 |- N = ( f e. ( C Func D ) , g e. ( C Func D ) |-> [_ ( 1st ` f ) / r ]_ [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` C ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` C ) A. y e. ( Base ` C ) A. h e. ( x ( Hom ` C ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } )
7		ovex 6678	. . . . . . 7 |- ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) e. _V
8	7	rgenw 2924	. . . . . 6 |- A. x e. ( Base ` C ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) e. _V
9		ixpexg 7932	. . . . . 6 |- ( A. x e. ( Base ` C ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) e. _V -> X_ x e. ( Base ` C ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) e. _V )
10	8, 9	ax-mp 5	. . . . 5 |- X_ x e. ( Base ` C ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) e. _V
11	10	rabex 4813	. . . 4 |- { a e. X_ x e. ( Base ` C ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` C ) A. y e. ( Base ` C ) A. h e. ( x ( Hom ` C ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } e. _V
12	11	csbex 4793	. . 3 |- [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` C ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` C ) A. y e. ( Base ` C ) A. h e. ( x ( Hom ` C ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } e. _V
13	12	csbex 4793	. 2 |- [_ ( 1st ` f ) / r ]_ [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` C ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` C ) A. y e. ( Base ` C ) A. h e. ( x ( Hom ` C ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } e. _V
14	6, 13	fnmpt2i 7239	1 |- N Fn ( ( C Func D ) X. ( C Func D ) )

Syntax hints:   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   [_csb 3533   <.cop 4183   X. cxp 5112   Fn wfn 5883   ` 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-fal 1489  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:  fuchom  16621