Theorem cofuval2 16547

Description: Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
cofuval2.b	|- B = ( Base ` C )
cofuval2.f	|- ( ph -> F ( C Func D ) G )
cofuval2.x	|- ( ph -> H ( D Func E ) K )

Assertion

Ref	Expression
cofuval2	|- ( ph -> ( <. H , K >. o.func <. F , G >. ) = <. ( H o. F ) , ( x e. B , y e. B |-> ( ( ( F ` x ) K ( F ` y ) ) o. ( x G y ) ) ) >. )

Distinct variable groups:   x, y, B   x, F, y   x, G, y   x, H, y   ph, x, y   x, K, y

Allowed substitution hints:   C( x, y)   D( x, y)   E( x, y)

Proof of Theorem cofuval2

Step	Hyp	Ref	Expression
1		cofuval2.b	. . 3 |- B = ( Base ` C )
2		cofuval2.f	. . . 4 |- ( ph -> F ( C Func D ) G )
3		df-br 4654	. . . 4 |- ( F ( C Func D ) G <-> <. F , G >. e. ( C Func D ) )
4	2, 3	sylib 208	. . 3 |- ( ph -> <. F , G >. e. ( C Func D ) )
5		cofuval2.x	. . . 4 |- ( ph -> H ( D Func E ) K )
6		df-br 4654	. . . 4 |- ( H ( D Func E ) K <-> <. H , K >. e. ( D Func E ) )
7	5, 6	sylib 208	. . 3 |- ( ph -> <. H , K >. e. ( D Func E ) )
8	1, 4, 7	cofuval 16542	. 2 |- ( ph -> ( <. H , K >. o.func <. F , G >. ) = <. ( ( 1st ` <. H , K >. ) o. ( 1st ` <. F , G >. ) ) , ( x e. B , y e. B |-> ( ( ( ( 1st ` <. F , G >. ) ` x ) ( 2nd ` <. H , K >. ) ( ( 1st ` <. F , G >. ) ` y ) ) o. ( x ( 2nd ` <. F , G >. ) y ) ) ) >. )
9		relfunc 16522	. . . . . 6 |- Rel ( D Func E )
10		brrelex12 5155	. . . . . 6 |- ( ( Rel ( D Func E ) /\ H ( D Func E ) K ) -> ( H e. _V /\ K e. _V ) )
11	9, 5, 10	sylancr 695	. . . . 5 |- ( ph -> ( H e. _V /\ K e. _V ) )
12		op1stg 7180	. . . . 5 |- ( ( H e. _V /\ K e. _V ) -> ( 1st ` <. H , K >. ) = H )
13	11, 12	syl 17	. . . 4 |- ( ph -> ( 1st ` <. H , K >. ) = H )
14		relfunc 16522	. . . . . 6 |- Rel ( C Func D )
15		brrelex12 5155	. . . . . 6 |- ( ( Rel ( C Func D ) /\ F ( C Func D ) G ) -> ( F e. _V /\ G e. _V ) )
16	14, 2, 15	sylancr 695	. . . . 5 |- ( ph -> ( F e. _V /\ G e. _V ) )
17		op1stg 7180	. . . . 5 |- ( ( F e. _V /\ G e. _V ) -> ( 1st ` <. F , G >. ) = F )
18	16, 17	syl 17	. . . 4 |- ( ph -> ( 1st ` <. F , G >. ) = F )
19	13, 18	coeq12d 5286	. . 3 |- ( ph -> ( ( 1st ` <. H , K >. ) o. ( 1st ` <. F , G >. ) ) = ( H o. F ) )
20		op2ndg 7181	. . . . . . . 8 |- ( ( H e. _V /\ K e. _V ) -> ( 2nd ` <. H , K >. ) = K )
21	11, 20	syl 17	. . . . . . 7 |- ( ph -> ( 2nd ` <. H , K >. ) = K )
22	21	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ x e. B /\ y e. B ) -> ( 2nd ` <. H , K >. ) = K )
23	18	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ x e. B /\ y e. B ) -> ( 1st ` <. F , G >. ) = F )
24	23	fveq1d 6193	. . . . . 6 |- ( ( ph /\ x e. B /\ y e. B ) -> ( ( 1st ` <. F , G >. ) ` x ) = ( F ` x ) )
25	23	fveq1d 6193	. . . . . 6 |- ( ( ph /\ x e. B /\ y e. B ) -> ( ( 1st ` <. F , G >. ) ` y ) = ( F ` y ) )
26	22, 24, 25	oveq123d 6671	. . . . 5 |- ( ( ph /\ x e. B /\ y e. B ) -> ( ( ( 1st ` <. F , G >. ) ` x ) ( 2nd ` <. H , K >. ) ( ( 1st ` <. F , G >. ) ` y ) ) = ( ( F ` x ) K ( F ` y ) ) )
27		op2ndg 7181	. . . . . . . 8 |- ( ( F e. _V /\ G e. _V ) -> ( 2nd ` <. F , G >. ) = G )
28	16, 27	syl 17	. . . . . . 7 |- ( ph -> ( 2nd ` <. F , G >. ) = G )
29	28	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ x e. B /\ y e. B ) -> ( 2nd ` <. F , G >. ) = G )
30	29	oveqd 6667	. . . . 5 |- ( ( ph /\ x e. B /\ y e. B ) -> ( x ( 2nd ` <. F , G >. ) y ) = ( x G y ) )
31	26, 30	coeq12d 5286	. . . 4 |- ( ( ph /\ x e. B /\ y e. B ) -> ( ( ( ( 1st ` <. F , G >. ) ` x ) ( 2nd ` <. H , K >. ) ( ( 1st ` <. F , G >. ) ` y ) ) o. ( x ( 2nd ` <. F , G >. ) y ) ) = ( ( ( F ` x ) K ( F ` y ) ) o. ( x G y ) ) )
32	31	mpt2eq3dva 6719	. . 3 |- ( ph -> ( x e. B , y e. B |-> ( ( ( ( 1st ` <. F , G >. ) ` x ) ( 2nd ` <. H , K >. ) ( ( 1st ` <. F , G >. ) ` y ) ) o. ( x ( 2nd ` <. F , G >. ) y ) ) ) = ( x e. B , y e. B |-> ( ( ( F ` x ) K ( F ` y ) ) o. ( x G y ) ) ) )
33	19, 32	opeq12d 4410	. 2 |- ( ph -> <. ( ( 1st ` <. H , K >. ) o. ( 1st ` <. F , G >. ) ) , ( x e. B , y e. B |-> ( ( ( ( 1st ` <. F , G >. ) ` x ) ( 2nd ` <. H , K >. ) ( ( 1st ` <. F , G >. ) ` y ) ) o. ( x ( 2nd ` <. F , G >. ) y ) ) ) >. = <. ( H o. F ) , ( x e. B , y e. B |-> ( ( ( F ` x ) K ( F ` y ) ) o. ( x G y ) ) ) >. )
34	8, 33	eqtrd 2656	1 |- ( ph -> ( <. H , K >. o.func <. F , G >. ) = <. ( H o. F ) , ( x e. B , y e. B |-> ( ( ( F ` x ) K ( F ` y ) ) o. ( x G y ) ) ) >. )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   class class class wbr 4653   o. ccom 5118   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Func cfunc 16514   o._func ccofu 16516

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-map 7859  df-ixp 7909  df-func 16518  df-cofu 16520

This theorem is referenced by:  catcisolem  16756  funcrngcsetcALT  41999