Theorem comfeqval 16368

Description: Equality of two compositions. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
comfeqval.b	|- B = ( Base ` C )
comfeqval.h	|- H = ( Hom ` C )
comfeqval.1	|- .x. = (comp ` C )
comfeqval.2	|- .xb = (comp ` D )
comfeqval.3	|- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )
comfeqval.4	|- ( ph -> (compf ` C ) = (compf ` D ) )
comfeqval.x	|- ( ph -> X e. B )
comfeqval.y	|- ( ph -> Y e. B )
comfeqval.z	|- ( ph -> Z e. B )
comfeqval.f	|- ( ph -> F e. ( X H Y ) )
comfeqval.g	|- ( ph -> G e. ( Y H Z ) )

Assertion

Ref	Expression
comfeqval	|- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) = ( G ( <. X , Y >. .xb Z ) F ) )

Proof of Theorem comfeqval

Step	Hyp	Ref	Expression
1		comfeqval.4	. . . 4 |- ( ph -> (compf ` C ) = (compf ` D ) )
2	1	oveqd 6667	. . 3 |- ( ph -> ( <. X , Y >. (compf ` C ) Z ) = ( <. X , Y >. (compf ` D ) Z ) )
3	2	oveqd 6667	. 2 |- ( ph -> ( G ( <. X , Y >. (compf ` C ) Z ) F ) = ( G ( <. X , Y >. (compf ` D ) Z ) F ) )
4		eqid 2622	. . 3 |- (compf ` C ) = (compf ` C )
5		comfeqval.b	. . 3 |- B = ( Base ` C )
6		comfeqval.h	. . 3 |- H = ( Hom ` C )
7		comfeqval.1	. . 3 |- .x. = (comp ` C )
8		comfeqval.x	. . 3 |- ( ph -> X e. B )
9		comfeqval.y	. . 3 |- ( ph -> Y e. B )
10		comfeqval.z	. . 3 |- ( ph -> Z e. B )
11		comfeqval.f	. . 3 |- ( ph -> F e. ( X H Y ) )
12		comfeqval.g	. . 3 |- ( ph -> G e. ( Y H Z ) )
13	4, 5, 6, 7, 8, 9, 10, 11, 12	comfval 16360	. 2 |- ( ph -> ( G ( <. X , Y >. (compf ` C ) Z ) F ) = ( G ( <. X , Y >. .x. Z ) F ) )
14		eqid 2622	. . 3 |- (compf ` D ) = (compf ` D )
15		eqid 2622	. . 3 |- ( Base ` D ) = ( Base ` D )
16		eqid 2622	. . 3 |- ( Hom ` D ) = ( Hom ` D )
17		comfeqval.2	. . 3 |- .xb = (comp ` D )
18		comfeqval.3	. . . . . 6 |- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )
19	18	homfeqbas 16356	. . . . 5 |- ( ph -> ( Base ` C ) = ( Base ` D ) )
20	5, 19	syl5eq 2668	. . . 4 |- ( ph -> B = ( Base ` D ) )
21	8, 20	eleqtrd 2703	. . 3 |- ( ph -> X e. ( Base ` D ) )
22	9, 20	eleqtrd 2703	. . 3 |- ( ph -> Y e. ( Base ` D ) )
23	10, 20	eleqtrd 2703	. . 3 |- ( ph -> Z e. ( Base ` D ) )
24	5, 6, 16, 18, 8, 9	homfeqval 16357	. . . 4 |- ( ph -> ( X H Y ) = ( X ( Hom ` D ) Y ) )
25	11, 24	eleqtrd 2703	. . 3 |- ( ph -> F e. ( X ( Hom ` D ) Y ) )
26	5, 6, 16, 18, 9, 10	homfeqval 16357	. . . 4 |- ( ph -> ( Y H Z ) = ( Y ( Hom ` D ) Z ) )
27	12, 26	eleqtrd 2703	. . 3 |- ( ph -> G e. ( Y ( Hom ` D ) Z ) )
28	14, 15, 16, 17, 21, 22, 23, 25, 27	comfval 16360	. 2 |- ( ph -> ( G ( <. X , Y >. (compf ` D ) Z ) F ) = ( G ( <. X , Y >. .xb Z ) F ) )
29	3, 13, 28	3eqtr3d 2664	1 |- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) = ( G ( <. X , Y >. .xb Z ) F ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   <.cop 4183   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952  compcco 15953   Hom _f chomf 16327  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-homf 16331  df-comf 16332

This theorem is referenced by:  catpropd  16369  cidpropd  16370  oppccomfpropd  16387  monpropd  16397  funcpropd  16560  natpropd  16636  fucpropd  16637  xpcpropd  16848  hofpropd  16907