Theorem cofuval 16542

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

Hypotheses

Ref	Expression
cofuval.b	|- B = ( Base ` C )
cofuval.f	|- ( ph -> F e. ( C Func D ) )
cofuval.g	|- ( ph -> G e. ( D Func E ) )

Assertion

Ref	Expression
cofuval	|- ( ph -> ( G o.func F ) = <. ( ( 1st ` G ) o. ( 1st ` F ) ) , ( x e. B , y e. B |-> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` G ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) ) >. )

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

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

Proof of Theorem cofuval

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cofu 16520	. . 3 |- o.func = ( g e. _V , f e. _V |-> <. ( ( 1st ` g ) o. ( 1st ` f ) ) , ( x e. dom dom ( 2nd ` f ) , y e. dom dom ( 2nd ` f ) |-> ( ( ( ( 1st ` f ) ` x ) ( 2nd ` g ) ( ( 1st ` f ) ` y ) ) o. ( x ( 2nd ` f ) y ) ) ) >. )
2	1	a1i 11	. 2 |- ( ph -> o.func = ( g e. _V , f e. _V |-> <. ( ( 1st ` g ) o. ( 1st ` f ) ) , ( x e. dom dom ( 2nd ` f ) , y e. dom dom ( 2nd ` f ) |-> ( ( ( ( 1st ` f ) ` x ) ( 2nd ` g ) ( ( 1st ` f ) ` y ) ) o. ( x ( 2nd ` f ) y ) ) ) >. ) )
3		simprl 794	. . . . 5 |- ( ( ph /\ ( g = G /\ f = F ) ) -> g = G )
4	3	fveq2d 6195	. . . 4 |- ( ( ph /\ ( g = G /\ f = F ) ) -> ( 1st ` g ) = ( 1st ` G ) )
5		simprr 796	. . . . 5 |- ( ( ph /\ ( g = G /\ f = F ) ) -> f = F )
6	5	fveq2d 6195	. . . 4 |- ( ( ph /\ ( g = G /\ f = F ) ) -> ( 1st ` f ) = ( 1st ` F ) )
7	4, 6	coeq12d 5286	. . 3 |- ( ( ph /\ ( g = G /\ f = F ) ) -> ( ( 1st ` g ) o. ( 1st ` f ) ) = ( ( 1st ` G ) o. ( 1st ` F ) ) )
8	5	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ ( g = G /\ f = F ) ) -> ( 2nd ` f ) = ( 2nd ` F ) )
9	8	dmeqd 5326	. . . . . . 7 |- ( ( ph /\ ( g = G /\ f = F ) ) -> dom ( 2nd ` f ) = dom ( 2nd ` F ) )
10		cofuval.b	. . . . . . . . . 10 |- B = ( Base ` C )
11		relfunc 16522	. . . . . . . . . . 11 |- Rel ( C Func D )
12		cofuval.f	. . . . . . . . . . 11 |- ( ph -> F e. ( C Func D ) )
13		1st2ndbr 7217	. . . . . . . . . . 11 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
14	11, 12, 13	sylancr 695	. . . . . . . . . 10 |- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
15	10, 14	funcfn2 16529	. . . . . . . . 9 |- ( ph -> ( 2nd ` F ) Fn ( B X. B ) )
16		fndm 5990	. . . . . . . . 9 |- ( ( 2nd ` F ) Fn ( B X. B ) -> dom ( 2nd ` F ) = ( B X. B ) )
17	15, 16	syl 17	. . . . . . . 8 |- ( ph -> dom ( 2nd ` F ) = ( B X. B ) )
18	17	adantr 481	. . . . . . 7 |- ( ( ph /\ ( g = G /\ f = F ) ) -> dom ( 2nd ` F ) = ( B X. B ) )
19	9, 18	eqtrd 2656	. . . . . 6 |- ( ( ph /\ ( g = G /\ f = F ) ) -> dom ( 2nd ` f ) = ( B X. B ) )
20	19	dmeqd 5326	. . . . 5 |- ( ( ph /\ ( g = G /\ f = F ) ) -> dom dom ( 2nd ` f ) = dom ( B X. B ) )
21		dmxpid 5345	. . . . 5 |- dom ( B X. B ) = B
22	20, 21	syl6eq 2672	. . . 4 |- ( ( ph /\ ( g = G /\ f = F ) ) -> dom dom ( 2nd ` f ) = B )
23	3	fveq2d 6195	. . . . . 6 |- ( ( ph /\ ( g = G /\ f = F ) ) -> ( 2nd ` g ) = ( 2nd ` G ) )
24	6	fveq1d 6193	. . . . . 6 |- ( ( ph /\ ( g = G /\ f = F ) ) -> ( ( 1st ` f ) ` x ) = ( ( 1st ` F ) ` x ) )
25	6	fveq1d 6193	. . . . . 6 |- ( ( ph /\ ( g = G /\ f = F ) ) -> ( ( 1st ` f ) ` y ) = ( ( 1st ` F ) ` y ) )
26	23, 24, 25	oveq123d 6671	. . . . 5 |- ( ( ph /\ ( g = G /\ f = F ) ) -> ( ( ( 1st ` f ) ` x ) ( 2nd ` g ) ( ( 1st ` f ) ` y ) ) = ( ( ( 1st ` F ) ` x ) ( 2nd ` G ) ( ( 1st ` F ) ` y ) ) )
27	8	oveqd 6667	. . . . 5 |- ( ( ph /\ ( g = G /\ f = F ) ) -> ( x ( 2nd ` f ) y ) = ( x ( 2nd ` F ) y ) )
28	26, 27	coeq12d 5286	. . . 4 |- ( ( ph /\ ( g = G /\ f = F ) ) -> ( ( ( ( 1st ` f ) ` x ) ( 2nd ` g ) ( ( 1st ` f ) ` y ) ) o. ( x ( 2nd ` f ) y ) ) = ( ( ( ( 1st ` F ) ` x ) ( 2nd ` G ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) )
29	22, 22, 28	mpt2eq123dv 6717	. . 3 |- ( ( ph /\ ( g = G /\ f = F ) ) -> ( x e. dom dom ( 2nd ` f ) , y e. dom dom ( 2nd ` f ) |-> ( ( ( ( 1st ` f ) ` x ) ( 2nd ` g ) ( ( 1st ` f ) ` y ) ) o. ( x ( 2nd ` f ) y ) ) ) = ( x e. B , y e. B |-> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` G ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) ) )
30	7, 29	opeq12d 4410	. 2 |- ( ( ph /\ ( g = G /\ f = F ) ) -> <. ( ( 1st ` g ) o. ( 1st ` f ) ) , ( x e. dom dom ( 2nd ` f ) , y e. dom dom ( 2nd ` f ) |-> ( ( ( ( 1st ` f ) ` x ) ( 2nd ` g ) ( ( 1st ` f ) ` y ) ) o. ( x ( 2nd ` f ) y ) ) ) >. = <. ( ( 1st ` G ) o. ( 1st ` F ) ) , ( x e. B , y e. B |-> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` G ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) ) >. )
31		cofuval.g	. . 3 |- ( ph -> G e. ( D Func E ) )
32		elex 3212	. . 3 |- ( G e. ( D Func E ) -> G e. _V )
33	31, 32	syl 17	. 2 |- ( ph -> G e. _V )
34		elex 3212	. . 3 |- ( F e. ( C Func D ) -> F e. _V )
35	12, 34	syl 17	. 2 |- ( ph -> F e. _V )
36		opex 4932	. . 3 |- <. ( ( 1st ` G ) o. ( 1st ` F ) ) , ( x e. B , y e. B |-> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` G ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) ) >. e. _V
37	36	a1i 11	. 2 |- ( ph -> <. ( ( 1st ` G ) o. ( 1st ` F ) ) , ( x e. B , y e. B |-> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` G ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) ) >. e. _V )
38	2, 30, 33, 35, 37	ovmpt2d 6788	1 |- ( ph -> ( G o.func F ) = <. ( ( 1st ` G ) o. ( 1st ` F ) ) , ( x e. B , y e. B |-> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` G ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) ) >. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   class class class wbr 4653   X. cxp 5112   dom cdm 5114   o. ccom 5118   Rel wrel 5119   Fn wfn 5883   ` 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:  cofu1st  16543  cofu2nd  16545  cofuval2  16547  cofucl  16548  cofuass  16549  cofulid  16550  cofurid  16551  prf1st  16844  prf2nd  16845