Theorem cofu2nd 16545

Description: Value of the morphism part of the functor composition. (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 ) )
cofu2nd.x	|- ( ph -> X e. B )
cofu2nd.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
cofu2nd	|- ( ph -> ( X ( 2nd ` ( G o.func F ) ) Y ) = ( ( ( ( 1st ` F ) ` X ) ( 2nd ` G ) ( ( 1st ` F ) ` Y ) ) o. ( X ( 2nd ` F ) Y ) ) )

Proof of Theorem cofu2nd

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cofuval.b	. . . . 5 |- B = ( Base ` C )
2		cofuval.f	. . . . 5 |- ( ph -> F e. ( C Func D ) )
3		cofuval.g	. . . . 5 |- ( ph -> G e. ( D Func E ) )
4	1, 2, 3	cofuval 16542	. . . 4 |- ( 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 ) ) ) >. )
5	4	fveq2d 6195	. . 3 |- ( ph -> ( 2nd ` ( G o.func F ) ) = ( 2nd ` <. ( ( 1st ` G ) o. ( 1st ` F ) ) , ( x e. B , y e. B |-> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` G ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) ) >. ) )
6		fvex 6201	. . . . 5 |- ( 1st ` G ) e. _V
7		fvex 6201	. . . . 5 |- ( 1st ` F ) e. _V
8	6, 7	coex 7118	. . . 4 |- ( ( 1st ` G ) o. ( 1st ` F ) ) e. _V
9		fvex 6201	. . . . . 6 |- ( Base ` C ) e. _V
10	1, 9	eqeltri 2697	. . . . 5 |- B e. _V
11	10, 10	mpt2ex 7247	. . . 4 |- ( x e. B , y e. B |-> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` G ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) ) e. _V
12	8, 11	op2nd 7177	. . 3 |- ( 2nd ` <. ( ( 1st ` G ) o. ( 1st ` F ) ) , ( x e. B , y e. B |-> ( ( ( ( 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 ) ) )
13	5, 12	syl6eq 2672	. 2 |- ( ph -> ( 2nd ` ( G o.func F ) ) = ( x e. B , y e. B |-> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` G ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) ) )
14		simprl 794	. . . . 5 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> x = X )
15	14	fveq2d 6195	. . . 4 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( ( 1st ` F ) ` x ) = ( ( 1st ` F ) ` X ) )
16		simprr 796	. . . . 5 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> y = Y )
17	16	fveq2d 6195	. . . 4 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( ( 1st ` F ) ` y ) = ( ( 1st ` F ) ` Y ) )
18	15, 17	oveq12d 6668	. . 3 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( ( ( 1st ` F ) ` x ) ( 2nd ` G ) ( ( 1st ` F ) ` y ) ) = ( ( ( 1st ` F ) ` X ) ( 2nd ` G ) ( ( 1st ` F ) ` Y ) ) )
19	14, 16	oveq12d 6668	. . 3 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( x ( 2nd ` F ) y ) = ( X ( 2nd ` F ) Y ) )
20	18, 19	coeq12d 5286	. 2 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( ( ( ( 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 ) ) )
21		cofu2nd.x	. 2 |- ( ph -> X e. B )
22		cofu2nd.y	. 2 |- ( ph -> Y e. B )
23		ovex 6678	. . . 4 |- ( ( ( 1st ` F ) ` X ) ( 2nd ` G ) ( ( 1st ` F ) ` Y ) ) e. _V
24		ovex 6678	. . . 4 |- ( X ( 2nd ` F ) Y ) e. _V
25	23, 24	coex 7118	. . 3 |- ( ( ( ( 1st ` F ) ` X ) ( 2nd ` G ) ( ( 1st ` F ) ` Y ) ) o. ( X ( 2nd ` F ) Y ) ) e. _V
26	25	a1i 11	. 2 |- ( ph -> ( ( ( ( 1st ` F ) ` X ) ( 2nd ` G ) ( ( 1st ` F ) ` Y ) ) o. ( X ( 2nd ` F ) Y ) ) e. _V )
27	13, 20, 21, 22, 26	ovmpt2d 6788	1 |- ( ph -> ( X ( 2nd ` ( G o.func F ) ) Y ) = ( ( ( ( 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   o. ccom 5118   ` 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:  cofu2  16546  cofucl  16548  cofuass  16549  cofull  16594  cofth  16595  catciso  16757