Theorem fuccofval 16619

Description: Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
fucval.q	|- Q = ( C FuncCat D )
fucval.b	|- B = ( C Func D )
fucval.n	|- N = ( C Nat D )
fucval.a	|- A = ( Base ` C )
fucval.o	|- .x. = (comp ` D )
fucval.c	|- ( ph -> C e. Cat )
fucval.d	|- ( ph -> D e. Cat )
fuccofval.x	|- .xb = (comp ` Q )

Assertion

Ref	Expression
fuccofval	|- ( ph -> .xb = ( v e. ( B X. B ) , h e. B |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) )

Distinct variable groups:   v, h, B   a, b, f, g, h, v, x, ph   C, a, b, f, g, h, v, x   D, a, b, f, g, h, v, x

Allowed substitution hints:   A( x, v, f, g, h, a, b)   B( x, f, g, a, b)   Q( x, v, f, g, h, a, b)   .xb ( x, v, f, g, h, a, b)   .x. ( x, v, f, g, h, a, b)   N( x, v, f, g, h, a, b)

Proof of Theorem fuccofval

Step	Hyp	Ref	Expression
1		fucval.q	. . . 4 |- Q = ( C FuncCat D )
2		fucval.b	. . . 4 |- B = ( C Func D )
3		fucval.n	. . . 4 |- N = ( C Nat D )
4		fucval.a	. . . 4 |- A = ( Base ` C )
5		fucval.o	. . . 4 |- .x. = (comp ` D )
6		fucval.c	. . . 4 |- ( ph -> C e. Cat )
7		fucval.d	. . . 4 |- ( ph -> D e. Cat )
8		eqidd 2623	. . . 4 |- ( ph -> ( v e. ( B X. B ) , h e. B |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) = ( v e. ( B X. B ) , h e. B |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) )
9	1, 2, 3, 4, 5, 6, 7, 8	fucval 16618	. . 3 |- ( ph -> Q = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , N >. , <. (comp ` ndx ) , ( v e. ( B X. B ) , h e. B |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. } )
10	9	fveq2d 6195	. 2 |- ( ph -> (comp ` Q ) = (comp ` { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , N >. , <. (comp ` ndx ) , ( v e. ( B X. B ) , h e. B |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. } ) )
11		fuccofval.x	. 2 |- .xb = (comp ` Q )
12		ovex 6678	. . . . . 6 |- ( C Func D ) e. _V
13	2, 12	eqeltri 2697	. . . . 5 |- B e. _V
14	13, 13	xpex 6962	. . . 4 |- ( B X. B ) e. _V
15	14, 13	mpt2ex 7247	. . 3 |- ( v e. ( B X. B ) , h e. B |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) e. _V
16		catstr 16617	. . . 4 |- { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , N >. , <. (comp ` ndx ) , ( v e. ( B X. B ) , h e. B |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. } Struct <. 1 , ; 1 5 >.
17		ccoid 16077	. . . 4 |- comp = Slot (comp ` ndx )
18		snsstp3 4349	. . . 4 |- { <. (comp ` ndx ) , ( v e. ( B X. B ) , h e. B |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. } C_ { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , N >. , <. (comp ` ndx ) , ( v e. ( B X. B ) , h e. B |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. }
19	16, 17, 18	strfv 15907	. . 3 |- ( ( v e. ( B X. B ) , h e. B |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) e. _V -> ( v e. ( B X. B ) , h e. B |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) = (comp ` { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , N >. , <. (comp ` ndx ) , ( v e. ( B X. B ) , h e. B |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. } ) )
20	15, 19	ax-mp 5	. 2 |- ( v e. ( B X. B ) , h e. B |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) = (comp ` { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , N >. , <. (comp ` ndx ) , ( v e. ( B X. B ) , h e. B |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. } )
21	10, 11, 20	3eqtr4g 2681	1 |- ( ph -> .xb = ( v e. ( B X. B ) , h e. B |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. A |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. .x. ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   {ctp 4181   <.cop 4183   |-> cmpt 4729   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   1c1 9937   5c5 11073  ;cdc 11493   ndxcnx 15854   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Func cfunc 16514   Nat cnat 16601   FuncCat cfuc 16602

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-fuc 16604

This theorem is referenced by:  fucbas  16620  fuchom  16621  fucco  16622