Theorem coafval 16714

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 6191	. . . . . 6 |- ( c = C -> (Nat ` c ) = (Nat ` C ) )
3		coafval.a	. . . . . 6 |- A = (Nat ` C )
4	2, 3	syl6eqr 2674	. . . . 5 |- ( c = C -> (Nat ` c ) = A )
5	4	rabeqdv 3194	. . . . 5 |- ( c = C -> { h e. (Nat ` c ) | (coda ` h ) = (domA ` g ) } = { h e. A | (coda ` h ) = (domA ` g ) } )
6		fveq2 6191	. . . . . . . . 9 |- ( c = C -> (comp ` c ) = (comp ` C ) )
7		coafval.x	. . . . . . . . 9 |- .xb = (comp ` C )
8	6, 7	syl6eqr 2674	. . . . . . . 8 |- ( c = C -> (comp ` c ) = .xb )
9	8	oveqd 6667	. . . . . . 7 |- ( c = C -> ( <. (domA ` f ) , (domA ` g ) >. (comp ` c ) (coda ` g ) ) = ( <. (domA ` f ) , (domA ` g ) >. .xb (coda ` g ) ) )
10	9	oveqd 6667	. . . . . 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 4416	. . . . 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 6717	. . . 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 16706	. . . 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 6201	. . . . . 6 |- (Nat ` C ) e. _V
15	3, 14	eqeltri 2697	. . . . 5 |- A e. _V
16	15	rabex 4813	. . . . 5 |- { h e. A | (coda ` h ) = (domA ` g ) } e. _V
17	15, 16	mpt2ex 7247	. . . 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 6282	. . 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 5631	. . . . . . 7 |- dom compa C_ Cat
20	19	sseli 3599	. . . . . 6 |- ( C e. dom compa -> C e. Cat )
21	20	con3i 150	. . . . 5 |- ( -. C e. Cat -> -. C e. dom compa )
22		ndmfv 6218	. . . . 5 |- ( -. C e. dom compa -> (compa ` C ) = (/) )
23	21, 22	syl 17	. . . 4 |- ( -. C e. Cat -> (compa ` C ) = (/) )
24	3	arwrcl 16694	. . . . . . . 8 |- ( f e. A -> C e. Cat )
25	24	con3i 150	. . . . . . 7 |- ( -. C e. Cat -> -. f e. A )
26	25	eq0rdv 3979	. . . . . 6 |- ( -. C e. Cat -> A = (/) )
27		eqidd 2623	. . . . . 6 |- ( -. C e. Cat -> { h e. A | (coda ` h ) = (domA ` g ) } = { h e. A | (coda ` h ) = (domA ` g ) } )
28		eqidd 2623	. . . . . 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 6717	. . . . 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 6725	. . . . 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 2672	. . . 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 2659	. . 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 176	. 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 2644	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 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   (/)c0 3915   <.cop 4183   <.cotp 4185   dom cdm 5114   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   2ndc2nd 7167  compcco 15953   Catccat 16325  domAcdoma 16670  cod_accoda 16671  Natcarw 16672  comp_accoa 16704

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-ot 4186  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-arw 16677  df-coa 16706

This theorem is referenced by:  eldmcoa  16715  dmcoass  16716  coaval  16718  coapm  16721