Theorem evlf1 16860

Description: Value of the evaluation functor at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
evlf1.e	|- E = ( C evalF D )
evlf1.c	|- ( ph -> C e. Cat )
evlf1.d	|- ( ph -> D e. Cat )
evlf1.b	|- B = ( Base ` C )
evlf1.f	|- ( ph -> F e. ( C Func D ) )
evlf1.x	|- ( ph -> X e. B )

Assertion

Ref	Expression
evlf1	|- ( ph -> ( F ( 1st ` E ) X ) = ( ( 1st ` F ) ` X ) )

Proof of Theorem evlf1

Dummy variables x y f a g m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evlf1.e	. . . 4 |- E = ( C evalF D )
2		evlf1.c	. . . 4 |- ( ph -> C e. Cat )
3		evlf1.d	. . . 4 |- ( ph -> D e. Cat )
4		evlf1.b	. . . 4 |- B = ( Base ` C )
5		eqid 2622	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
6		eqid 2622	. . . 4 |- (comp ` D ) = (comp ` D )
7		eqid 2622	. . . 4 |- ( C Nat D ) = ( C Nat D )
8	1, 2, 3, 4, 5, 6, 7	evlfval 16857	. . 3 |- ( ph -> E = <. ( f e. ( C Func D ) , x e. B |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( C Func D ) X. B ) , y e. ( ( C Func D ) X. B ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. )
9		ovex 6678	. . . . 5 |- ( C Func D ) e. _V
10		fvex 6201	. . . . . 6 |- ( Base ` C ) e. _V
11	4, 10	eqeltri 2697	. . . . 5 |- B e. _V
12	9, 11	mpt2ex 7247	. . . 4 |- ( f e. ( C Func D ) , x e. B |-> ( ( 1st ` f ) ` x ) ) e. _V
13	9, 11	xpex 6962	. . . . 5 |- ( ( C Func D ) X. B ) e. _V
14	13, 13	mpt2ex 7247	. . . 4 |- ( x e. ( ( C Func D ) X. B ) , y e. ( ( C Func D ) X. B ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) e. _V
15	12, 14	op1std 7178	. . 3 |- ( E = <. ( f e. ( C Func D ) , x e. B |-> ( ( 1st ` f ) ` x ) ) , ( x e. ( ( C Func D ) X. B ) , y e. ( ( C Func D ) X. B ) |-> [_ ( 1st ` x ) / m ]_ [_ ( 1st ` y ) / n ]_ ( a e. ( m ( C Nat D ) n ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( ( a ` ( 2nd ` y ) ) ( <. ( ( 1st ` m ) ` ( 2nd ` x ) ) , ( ( 1st ` m ) ` ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` n ) ` ( 2nd ` y ) ) ) ( ( ( 2nd ` x ) ( 2nd ` m ) ( 2nd ` y ) ) ` g ) ) ) ) >. -> ( 1st ` E ) = ( f e. ( C Func D ) , x e. B |-> ( ( 1st ` f ) ` x ) ) )
16	8, 15	syl 17	. 2 |- ( ph -> ( 1st ` E ) = ( f e. ( C Func D ) , x e. B |-> ( ( 1st ` f ) ` x ) ) )
17		simprl 794	. . . 4 |- ( ( ph /\ ( f = F /\ x = X ) ) -> f = F )
18	17	fveq2d 6195	. . 3 |- ( ( ph /\ ( f = F /\ x = X ) ) -> ( 1st ` f ) = ( 1st ` F ) )
19		simprr 796	. . 3 |- ( ( ph /\ ( f = F /\ x = X ) ) -> x = X )
20	18, 19	fveq12d 6197	. 2 |- ( ( ph /\ ( f = F /\ x = X ) ) -> ( ( 1st ` f ) ` x ) = ( ( 1st ` F ) ` X ) )
21		evlf1.f	. 2 |- ( ph -> F e. ( C Func D ) )
22		evlf1.x	. 2 |- ( ph -> X e. B )
23		fvexd 6203	. 2 |- ( ph -> ( ( 1st ` F ) ` X ) e. _V )
24	16, 20, 21, 22, 23	ovmpt2d 6788	1 |- ( ph -> ( F ( 1st ` E ) X ) = ( ( 1st ` F ) ` X ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   <.cop 4183   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Func cfunc 16514   Nat cnat 16601   eval_F cevlf 16849

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-evlf 16853

This theorem is referenced by:  evlfcllem  16861  evlfcl  16862  uncf1  16876  yonedalem3a  16914  yonedalem3b  16919  yonedainv  16921  yonffthlem  16922