Theorem coafval 15470

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 2516	. . . . 5 |- ( c = C -> (Nat ` c ) = A )
5		biidd 237	. . . . . 6 |- ( c = C -> ( (coda ` h ) = (domA ` g ) <-> (coda ` h ) = (domA ` g ) ) )
6	4, 5	rabeqbidv 3104	. . . . 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 2516	. . . . . . . 8 |- ( c = C -> (comp ` c ) = .xb )
10	9	oveqd 6313	. . . . . . 7 |- ( c = C -> ( <. (domA ` f ) , (domA ` g ) >. (comp ` c ) (coda ` g ) ) = ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) )
11	10	oveqd 6313	. . . . . 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 6358	. . . 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 15462	. . . 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 2541	. . . . 5 |- A e. _V
17	16	rabex 4607	. . . . 5 |- { h e. A | (coda ` h ) = (domA ` g ) } e. _V
18	16, 17	mpt2ex 6876	. . . 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 5956	. . 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 3495	. . . . . 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 15450	. . . . . . . 8 |- ( f e. A -> C e. Cat )
26	25	con3i 135	. . . . . . 7 |- ( -. C e. Cat -> -. f e. A )
27	26	eq0rdv 3829	. . . . . 6 |- ( -. C e. Cat -> A = (/) )
28		eqidd 2458	. . . . . 6 |- ( -. C e. Cat -> { h e. A | (coda ` h ) = (domA ` g ) } = { h e. A | (coda ` h ) = (domA ` g ) } )
29		eqidd 2458	. . . . . 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 6358	. . . . 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 6366	. . . . 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 2514	. . . 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 2501	. . 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 2486	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 1395   e. wcel 1819   {crab 2811   _Vcvv 3109   (/)c0 3793   <.cop 4038   <.cotp 4040   dom cdm 5008   ` cfv 5594  (class class class)co 6296   |-> cmpt2 6298   2ndc2nd 6798  compcco 14724   Catccat 15081  domAcdoma 15426  cod_accoda 15427  Natcarw 15428  comp_accoa 15460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-ot 4041  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-arw 15433  df-coa 15462

This theorem is referenced by:  eldmcoa  15471  dmcoass  15472  coaval  15474  coapm  15477