Theorem coafval 16037

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 5879	. . . . . 6 |- ( c = C -> (Nat ` c ) = (Nat ` C ) )
3		coafval.a	. . . . . 6 |- A = (Nat ` C )
4	2, 3	syl6eqr 2523	. . . . 5 |- ( c = C -> (Nat ` c ) = A )
5	4	rabeqdv 3025	. . . . 5 |- ( c = C -> { h e. (Nat ` c ) | (coda ` h ) = (domA ` g ) } = { h e. A | (coda ` h ) = (domA ` g ) } )
6		fveq2 5879	. . . . . . . . 9 |- ( c = C -> (comp ` c ) = (comp ` C ) )
7		coafval.x	. . . . . . . . 9 |- .xb = (comp ` C )
8	6, 7	syl6eqr 2523	. . . . . . . 8 |- ( c = C -> (comp ` c ) = .xb )
9	8	oveqd 6325	. . . . . . 7 |- ( c = C -> ( <. (domA ` f ) , (domA ` g ) >. (comp ` c ) (coda ` g ) ) = ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) )
10	9	oveqd 6325	. . . . . 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 ) ) )
11	10	oteq3d 4172	. . . . 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 ) ) >. )
12	4, 5, 11	mpt2eq123dv 6372	. . . 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 ) ) >. ) )
13		df-coa 16029	. . . 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 ) ) >. ) )
14		fvex 5889	. . . . . 6 |- (Nat ` C ) e. _V
15	3, 14	eqeltri 2545	. . . . 5 |- A e. _V
16	15	rabex 4550	. . . . 5 |- { h e. A | (coda ` h ) = (domA ` g ) } e. _V
17	15, 16	mpt2ex 6889	. . . 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
18	12, 13, 17	fvmpt 5963	. . 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 ) ) >. ) )
19	13	dmmptss 5338	. . . . . . 7 |- dom compa C_ Cat
20	19	sseli 3414	. . . . . 6 |- ( C e. dom compa -> C e. Cat )
21	20	con3i 142	. . . . 5 |- ( -. C e. Cat -> -. C e. dom compa )
22		ndmfv 5903	. . . . 5 |- ( -. C e. dom compa -> (compa ` C ) = (/) )
23	21, 22	syl 17	. . . 4 |- ( -. C e. Cat -> (compa ` C ) = (/) )
24	3	arwrcl 16017	. . . . . . . 8 |- ( f e. A -> C e. Cat )
25	24	con3i 142	. . . . . . 7 |- ( -. C e. Cat -> -. f e. A )
26	25	eq0rdv 3773	. . . . . 6 |- ( -. C e. Cat -> A = (/) )
27		eqidd 2472	. . . . . 6 |- ( -. C e. Cat -> { h e. A | (coda ` h ) = (domA ` g ) } = { h e. A | (coda ` h ) = (domA ` g ) } )
28		eqidd 2472	. . . . . 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 ) ) >. )
29	26, 27, 28	mpt2eq123dv 6372	. . . . 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 ) ) >. ) )
30		mpt20 6380	. . . . 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 ) ) >. ) = (/)
31	29, 30	syl6eq 2521	. . . 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 ) ) >. ) = (/) )
32	23, 31	eqtr4d 2508	. . 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 ) ) >. ) )
33	18, 32	pm2.61i 169	. 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 ) ) >. )
34	1, 33	eqtri 2493	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 1452   e. wcel 1904   {crab 2760   _Vcvv 3031   (/)c0 3722   <.cop 3965   <.cotp 3967   dom cdm 4839   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310   2ndc2nd 6811  compcco 15280   Catccat 15648  domAcdoma 15993  cod_accoda 15994  Natcarw 15995  comp_accoa 16027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-ot 3968  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-arw 16000  df-coa 16029

This theorem is referenced by:  eldmcoa  16038  dmcoass  16039  coaval  16041  coapm  16044