Theorem coafval 14947

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 5706	. . . . . 6 |- ( c = C -> (Nat ` c ) = (Nat ` C ) )
3		coafval.a	. . . . . 6 |- A = (Nat ` C )
4	2, 3	syl6eqr 2493	. . . . 5 |- ( c = C -> (Nat ` c ) = A )
5		biidd 237	. . . . . 6 |- ( c = C -> ( (coda ` h ) = (domA ` g ) <-> (coda ` h ) = (domA ` g ) ) )
6	4, 5	rabeqbidv 2982	. . . . 5 |- ( c = C -> { h e. (Nat ` c ) | (coda ` h ) = (domA ` g ) } = { h e. A | (coda ` h ) = (domA ` g ) } )
7		fveq2 5706	. . . . . . . . 9 |- ( c = C -> (comp ` c ) = (comp ` C ) )
8		coafval.x	. . . . . . . . 9 |- .xb = (comp ` C )
9	7, 8	syl6eqr 2493	. . . . . . . 8 |- ( c = C -> (comp ` c ) = .xb )
10	9	oveqd 6123	. . . . . . 7 |- ( c = C -> ( <. (domA ` f ) , (domA ` g ) >. (comp ` c ) (coda ` g ) ) = ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) )
11	10	oveqd 6123	. . . . . 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 4088	. . . . 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 6163	. . . 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 14939	. . . 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 5716	. . . . . 6 |- (Nat ` C ) e. _V
16	3, 15	eqeltri 2513	. . . . 5 |- A e. _V
17	16	rabex 4458	. . . . 5 |- { h e. A | (coda ` h ) = (domA ` g ) } e. _V
18	16, 17	mpt2ex 6665	. . . 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 5789	. . 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 5349	. . . . . . 7 |- dom compa C_ Cat
21	20	sseli 3367	. . . . . 6 |- ( C e. dom compa -> C e. Cat )
22	21	con3i 135	. . . . 5 |- ( -. C e. Cat -> -. C e. dom compa )
23		ndmfv 5729	. . . . 5 |- ( -. C e. dom compa -> (compa ` C ) = (/) )
24	22, 23	syl 16	. . . 4 |- ( -. C e. Cat -> (compa ` C ) = (/) )
25	3	arwrcl 14927	. . . . . . . 8 |- ( f e. A -> C e. Cat )
26	25	con3i 135	. . . . . . 7 |- ( -. C e. Cat -> -. f e. A )
27	26	eq0rdv 3687	. . . . . 6 |- ( -. C e. Cat -> A = (/) )
28		eqidd 2444	. . . . . 6 |- ( -. C e. Cat -> { h e. A | (coda ` h ) = (domA ` g ) } = { h e. A | (coda ` h ) = (domA ` g ) } )
29		eqidd 2444	. . . . . 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 6163	. . . . 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 6171	. . . . 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 2491	. . . 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 2478	. . 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 2463	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 1369   e. wcel 1756   {crab 2734   _Vcvv 2987   (/)c0 3652   <.cop 3898   <.cotp 3900   dom cdm 4855   ` cfv 5433  (class class class)co 6106   e. cmpt2 6108   2ndc2nd 6591  compcco 14265   Catccat 14617  domAcdoma 14903  cod_accoda 14904  Natcarw 14905  comp_accoa 14937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-ot 3901  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-1st 6592  df-2nd 6593  df-arw 14910  df-coa 14939

This theorem is referenced by:  eldmcoa  14948  dmcoass  14949  coaval  14951  coapm  14954