Theorem coafval 15266

Description: The value of the composition of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
coafval.o	|- .x. = (compa ` C )
coafval.a	|- A = (Nat ` C )
coafval.x	|- .xb = (comp ` C )

Assertion

Ref	Expression
coafval	|- .x. = ( g e. A , f e. { h e. A | (coda ` h ) = (domA ` g ) } |-> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. )

Distinct variable groups:   f, g, h, A   C, f, g, h

Allowed substitution hints:   .xb ( f, g, h)   .x. ( f, g, h)

Proof of Theorem coafval

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		coafval.o	. 2 |- .x. = (compa ` C )
2		fveq2 5872	. . . . . 6 |- ( c = C -> (Nat ` c ) = (Nat ` C ) )
3		coafval.a	. . . . . 6 |- A = (Nat ` C )
4	2, 3	syl6eqr 2526	. . . . 5 |- ( c = C -> (Nat ` c ) = A )
5		biidd 237	. . . . . 6 |- ( c = C -> ( (coda ` h ) = (domA ` g ) <-> (coda ` h ) = (domA ` g ) ) )
6	4, 5	rabeqbidv 3113	. . . . 5 |- ( c = C -> { h e. (Nat ` c ) | (coda ` h ) = (domA ` g ) } = { h e. A | (coda ` h ) = (domA ` g ) } )
7		fveq2 5872	. . . . . . . . 9 |- ( c = C -> (comp ` c ) = (comp ` C ) )
8		coafval.x	. . . . . . . . 9 |- .xb = (comp ` C )
9	7, 8	syl6eqr 2526	. . . . . . . 8 |- ( c = C -> (comp ` c ) = .xb )
10	9	oveqd 6312	. . . . . . 7 |- ( c = C -> ( <. (domA ` f ) , (domA ` g ) >. (comp ` c ) (coda ` g ) ) = ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) )
11	10	oveqd 6312	. . . . . 6 |- ( c = C -> ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. (comp ` c ) (coda ` g ) ) ( 2nd ` f ) ) = ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) )
12	11	oteq3d 4233	. . . . 5 |- ( c = C -> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. (comp ` c ) (coda ` g ) ) ( 2nd ` f ) ) >. = <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. )
13	4, 6, 12	mpt2eq123dv 6354	. . . 4 |- ( c = C -> ( g e. (Nat ` c ) , f e. { h e. (Nat ` c ) | (coda ` h ) = (domA ` g ) } |-> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. (comp ` c ) (coda ` g ) ) ( 2nd ` f ) ) >. ) = ( g e. A , f e. { h e. A | (coda ` h ) = (domA ` g ) } |-> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. ) )
14		df-coa 15258	. . . 4 |- compa = ( c e. Cat |-> ( g e. (Nat ` c ) , f e. { h e. (Nat ` c ) | (coda ` h ) = (domA ` g ) } |-> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. (comp ` c ) (coda ` g ) ) ( 2nd ` f ) ) >. ) )
15		fvex 5882	. . . . . 6 |- (Nat ` C ) e. _V
16	3, 15	eqeltri 2551	. . . . 5 |- A e. _V
17	16	rabex 4604	. . . . 5 |- { h e. A | (coda ` h ) = (domA ` g ) } e. _V
18	16, 17	mpt2ex 6872	. . . 4 |- ( g e. A , f e. { h e. A | (coda ` h ) = (domA ` g ) } |-> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. ) e. _V
19	13, 14, 18	fvmpt 5957	. . 3 |- ( C e. Cat -> (compa ` C ) = ( g e. A , f e. { h e. A | (coda ` h ) = (domA ` g ) } |-> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. ) )
20	14	dmmptss 5509	. . . . . . 7 |- dom compa C_ Cat
21	20	sseli 3505	. . . . . 6 |- ( C e. dom compa -> C e. Cat )
22	21	con3i 135	. . . . 5 |- ( -. C e. Cat -> -. C e. dom compa )
23		ndmfv 5896	. . . . 5 |- ( -. C e. dom compa -> (compa ` C ) = (/) )
24	22, 23	syl 16	. . . 4 |- ( -. C e. Cat -> (compa ` C ) = (/) )
25	3	arwrcl 15246	. . . . . . . 8 |- ( f e. A -> C e. Cat )
26	25	con3i 135	. . . . . . 7 |- ( -. C e. Cat -> -. f e. A )
27	26	eq0rdv 3825	. . . . . 6 |- ( -. C e. Cat -> A = (/) )
28		eqidd 2468	. . . . . 6 |- ( -. C e. Cat -> { h e. A | (coda ` h ) = (domA ` g ) } = { h e. A | (coda ` h ) = (domA ` g ) } )
29		eqidd 2468	. . . . . 6 |- ( -. C e. Cat -> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. = <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. )
30	27, 28, 29	mpt2eq123dv 6354	. . . . 5 |- ( -. C e. Cat -> ( g e. A , f e. { h e. A | (coda ` h ) = (domA ` g ) } |-> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. ) = ( g e. (/) , f e. { h e. A | (coda ` h ) = (domA ` g ) } |-> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. ) )
31		mpt20 6362	. . . . 5 |- ( g e. (/) , f e. { h e. A | (coda ` h ) = (domA ` g ) } |-> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. ) = (/)
32	30, 31	syl6eq 2524	. . . 4 |- ( -. C e. Cat -> ( g e. A , f e. { h e. A | (coda ` h ) = (domA ` g ) } |-> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. ) = (/) )
33	24, 32	eqtr4d 2511	. . 3 |- ( -. C e. Cat -> (compa ` C ) = ( g e. A , f e. { h e. A | (coda ` h ) = (domA ` g ) } |-> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. ) )
34	19, 33	pm2.61i 164	. 2 |- (compa ` C ) = ( g e. A , f e. { h e. A | (coda ` h ) = (domA ` g ) } |-> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. )
35	1, 34	eqtri 2496	1 |- .x. = ( g e. A , f e. { h e. A | (coda ` h ) = (domA ` g ) } |-> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) ( 2nd ` f ) ) >. )

Syntax hints:   -. wn 3   = wceq 1379   e. wcel 1767   {crab 2821   _Vcvv 3118   (/)c0 3790   <.cop 4039   <.cotp 4041   dom cdm 5005   ` cfv 5594  (class class class)co 6295   |-> cmpt2 6297   2ndc2nd 6794  compcco 14584   Catccat 14936  domAcdoma 15222  cod_accoda 15223  Natcarw 15224  comp_accoa 15256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-ot 4042  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-arw 15229  df-coa 15258

This theorem is referenced by:  eldmcoa  15267  dmcoass  15268  coaval  15270  coapm  15273