Theorem hof2val 16896

Description: The morphism part of the Hom functor, for morphisms <. f , g >. : <. X , Y >. --> <. Z , W >. (which since the first argument is contravariant means morphisms f : Z --> X and g : Y --> W), yields a function (a morphism of SetCat) mapping h : X --> Y to g o. h o. f : Z --> W. (Contributed by Mario Carneiro, 15-Jan-2017.)

Hypotheses

Ref	Expression
hofval.m	|- M = (HomF ` C )
hofval.c	|- ( ph -> C e. Cat )
hof1.b	|- B = ( Base ` C )
hof1.h	|- H = ( Hom ` C )
hof1.x	|- ( ph -> X e. B )
hof1.y	|- ( ph -> Y e. B )
hof2.z	|- ( ph -> Z e. B )
hof2.w	|- ( ph -> W e. B )
hof2.o	|- .x. = (comp ` C )
hof2.f	|- ( ph -> F e. ( Z H X ) )
hof2.g	|- ( ph -> G e. ( Y H W ) )

Assertion

Ref	Expression
hof2val	|- ( ph -> ( F ( <. X , Y >. ( 2nd ` M ) <. Z , W >. ) G ) = ( h e. ( X H Y ) |-> ( ( G ( <. X , Y >. .x. W ) h ) ( <. Z , X >. .x. W ) F ) ) )

Distinct variable groups:   B, h   h, F   h, G   ph, h   C, h   h, H   h, W   .x. , h   h, X   h, Y   h, Z

Allowed substitution hint:   M( h)

Proof of Theorem hof2val

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

Step	Hyp	Ref	Expression
1		hofval.m	. . 3 |- M = (HomF ` C )
2		hofval.c	. . 3 |- ( ph -> C e. Cat )
3		hof1.b	. . 3 |- B = ( Base ` C )
4		hof1.h	. . 3 |- H = ( Hom ` C )
5		hof1.x	. . 3 |- ( ph -> X e. B )
6		hof1.y	. . 3 |- ( ph -> Y e. B )
7		hof2.z	. . 3 |- ( ph -> Z e. B )
8		hof2.w	. . 3 |- ( ph -> W e. B )
9		hof2.o	. . 3 |- .x. = (comp ` C )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	hof2fval 16895	. 2 |- ( ph -> ( <. X , Y >. ( 2nd ` M ) <. Z , W >. ) = ( f e. ( Z H X ) , g e. ( Y H W ) |-> ( h e. ( X H Y ) |-> ( ( g ( <. X , Y >. .x. W ) h ) ( <. Z , X >. .x. W ) f ) ) ) )
11		simplrr 801	. . . . 5 |- ( ( ( ph /\ ( f = F /\ g = G ) ) /\ h e. ( X H Y ) ) -> g = G )
12	11	oveq1d 6665	. . . 4 |- ( ( ( ph /\ ( f = F /\ g = G ) ) /\ h e. ( X H Y ) ) -> ( g ( <. X , Y >. .x. W ) h ) = ( G ( <. X , Y >. .x. W ) h ) )
13		simplrl 800	. . . 4 |- ( ( ( ph /\ ( f = F /\ g = G ) ) /\ h e. ( X H Y ) ) -> f = F )
14	12, 13	oveq12d 6668	. . 3 |- ( ( ( ph /\ ( f = F /\ g = G ) ) /\ h e. ( X H Y ) ) -> ( ( g ( <. X , Y >. .x. W ) h ) ( <. Z , X >. .x. W ) f ) = ( ( G ( <. X , Y >. .x. W ) h ) ( <. Z , X >. .x. W ) F ) )
15	14	mpteq2dva 4744	. 2 |- ( ( ph /\ ( f = F /\ g = G ) ) -> ( h e. ( X H Y ) |-> ( ( g ( <. X , Y >. .x. W ) h ) ( <. Z , X >. .x. W ) f ) ) = ( h e. ( X H Y ) |-> ( ( G ( <. X , Y >. .x. W ) h ) ( <. Z , X >. .x. W ) F ) ) )
16		hof2.f	. 2 |- ( ph -> F e. ( Z H X ) )
17		hof2.g	. 2 |- ( ph -> G e. ( Y H W ) )
18		ovex 6678	. . . 4 |- ( X H Y ) e. _V
19	18	mptex 6486	. . 3 |- ( h e. ( X H Y ) |-> ( ( G ( <. X , Y >. .x. W ) h ) ( <. Z , X >. .x. W ) F ) ) e. _V
20	19	a1i 11	. 2 |- ( ph -> ( h e. ( X H Y ) |-> ( ( G ( <. X , Y >. .x. W ) h ) ( <. Z , X >. .x. W ) F ) ) e. _V )
21	10, 15, 16, 17, 20	ovmpt2d 6788	1 |- ( ph -> ( F ( <. X , Y >. ( 2nd ` M ) <. Z , W >. ) G ) = ( h e. ( X H Y ) |-> ( ( G ( <. X , Y >. .x. W ) h ) ( <. Z , X >. .x. W ) F ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   2ndc2nd 7167   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325  Hom_Fchof 16888

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-hof 16890

This theorem is referenced by:  hof2  16897  hofcllem  16898  hofcl  16899  yonedalem3b  16919