Theorem comffn 16365

Description: The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
comfffn.o	|- O = (compf ` C )
comfffn.b	|- B = ( Base ` C )
comffn.h	|- H = ( Hom ` C )
comffn.x	|- ( ph -> X e. B )
comffn.y	|- ( ph -> Y e. B )
comffn.z	|- ( ph -> Z e. B )

Assertion

Ref	Expression
comffn	|- ( ph -> ( <. X , Y >. O Z ) Fn ( ( Y H Z ) X. ( X H Y ) ) )

Proof of Theorem comffn

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( g e. ( Y H Z ) , f e. ( X H Y ) |-> ( g ( <. X , Y >. (comp ` C ) Z ) f ) ) = ( g e. ( Y H Z ) , f e. ( X H Y ) |-> ( g ( <. X , Y >. (comp ` C ) Z ) f ) )
2		ovex 6678	. . 3 |- ( g ( <. X , Y >. (comp ` C ) Z ) f ) e. _V
3	1, 2	fnmpt2i 7239	. 2 |- ( g e. ( Y H Z ) , f e. ( X H Y ) |-> ( g ( <. X , Y >. (comp ` C ) Z ) f ) ) Fn ( ( Y H Z ) X. ( X H Y ) )
4		comfffn.o	. . . 4 |- O = (compf ` C )
5		comfffn.b	. . . 4 |- B = ( Base ` C )
6		comffn.h	. . . 4 |- H = ( Hom ` C )
7		eqid 2622	. . . 4 |- (comp ` C ) = (comp ` C )
8		comffn.x	. . . 4 |- ( ph -> X e. B )
9		comffn.y	. . . 4 |- ( ph -> Y e. B )
10		comffn.z	. . . 4 |- ( ph -> Z e. B )
11	4, 5, 6, 7, 8, 9, 10	comffval 16359	. . 3 |- ( ph -> ( <. X , Y >. O Z ) = ( g e. ( Y H Z ) , f e. ( X H Y ) |-> ( g ( <. X , Y >. (comp ` C ) Z ) f ) ) )
12	11	fneq1d 5981	. 2 |- ( ph -> ( ( <. X , Y >. O Z ) Fn ( ( Y H Z ) X. ( X H Y ) ) <-> ( g e. ( Y H Z ) , f e. ( X H Y ) |-> ( g ( <. X , Y >. (comp ` C ) Z ) f ) ) Fn ( ( Y H Z ) X. ( X H Y ) ) ) )
13	3, 12	mpbiri 248	1 |- ( ph -> ( <. X , Y >. O Z ) Fn ( ( Y H Z ) X. ( X H Y ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   <.cop 4183   X. cxp 5112   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   Hom chom 15952  compcco 15953  comp_fccomf 16328

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-comf 16332

This theorem is referenced by: (None)