Theorem coapm 16721

Description: Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

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

Assertion

Ref	Expression
coapm	|- .x. e. ( A ^pm ( A X. A ) )

Proof of Theorem coapm

Dummy variables f g h z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		coapm.o	. . . . . 6 |- .x. = (compa ` C )
2		coapm.a	. . . . . 6 |- A = (Nat ` C )
3		eqid 2622	. . . . . 6 |- (comp ` C ) = (comp ` C )
4	1, 2, 3	coafval 16714	. . . . 5 |- .x. = ( g e. A , f e. { h e. A | (coda ` h ) = (domA ` g ) } |-> <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. (comp ` C ) (coda ` g ) ) ( 2nd ` f ) ) >. )
5	4	mpt2fun 6762	. . . 4 |- Fun .x.
6		funfn 5918	. . . 4 |- ( Fun .x. <-> .x. Fn dom .x. )
7	5, 6	mpbi 220	. . 3 |- .x. Fn dom .x.
8	1, 2	dmcoass 16716	. . . . . . . . 9 |- dom .x. C_ ( A X. A )
9	8	sseli 3599	. . . . . . . 8 |- ( z e. dom .x. -> z e. ( A X. A ) )
10		1st2nd2 7205	. . . . . . . 8 |- ( z e. ( A X. A ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
11	9, 10	syl 17	. . . . . . 7 |- ( z e. dom .x. -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
12	11	fveq2d 6195	. . . . . 6 |- ( z e. dom .x. -> ( .x. ` z ) = ( .x. ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
13		df-ov 6653	. . . . . 6 |- ( ( 1st ` z ) .x. ( 2nd ` z ) ) = ( .x. ` <. ( 1st ` z ) , ( 2nd ` z ) >. )
14	12, 13	syl6eqr 2674	. . . . 5 |- ( z e. dom .x. -> ( .x. ` z ) = ( ( 1st ` z ) .x. ( 2nd ` z ) ) )
15		eqid 2622	. . . . . . 7 |- (Homa ` C ) = (Homa ` C )
16	2, 15	homarw 16696	. . . . . 6 |- ( (domA ` ( 2nd ` z ) ) (Homa ` C ) (coda ` ( 1st ` z ) ) ) C_ A
17		id 22	. . . . . . . . . . . . 13 |- ( z e. dom .x. -> z e. dom .x. )
18	11, 17	eqeltrrd 2702	. . . . . . . . . . . 12 |- ( z e. dom .x. -> <. ( 1st ` z ) , ( 2nd ` z ) >. e. dom .x. )
19		df-br 4654	. . . . . . . . . . . 12 |- ( ( 1st ` z ) dom .x. ( 2nd ` z ) <-> <. ( 1st ` z ) , ( 2nd ` z ) >. e. dom .x. )
20	18, 19	sylibr 224	. . . . . . . . . . 11 |- ( z e. dom .x. -> ( 1st ` z ) dom .x. ( 2nd ` z ) )
21	1, 2	eldmcoa 16715	. . . . . . . . . . 11 |- ( ( 1st ` z ) dom .x. ( 2nd ` z ) <-> ( ( 2nd ` z ) e. A /\ ( 1st ` z ) e. A /\ (coda ` ( 2nd ` z ) ) = (domA ` ( 1st ` z ) ) ) )
22	20, 21	sylib 208	. . . . . . . . . 10 |- ( z e. dom .x. -> ( ( 2nd ` z ) e. A /\ ( 1st ` z ) e. A /\ (coda ` ( 2nd ` z ) ) = (domA ` ( 1st ` z ) ) ) )
23	22	simp1d 1073	. . . . . . . . 9 |- ( z e. dom .x. -> ( 2nd ` z ) e. A )
24	2, 15	arwhoma 16695	. . . . . . . . 9 |- ( ( 2nd ` z ) e. A -> ( 2nd ` z ) e. ( (domA ` ( 2nd ` z ) ) (Homa ` C ) (coda ` ( 2nd ` z ) ) ) )
25	23, 24	syl 17	. . . . . . . 8 |- ( z e. dom .x. -> ( 2nd ` z ) e. ( (domA ` ( 2nd ` z ) ) (Homa ` C ) (coda ` ( 2nd ` z ) ) ) )
26	22	simp3d 1075	. . . . . . . . 9 |- ( z e. dom .x. -> (coda ` ( 2nd ` z ) ) = (domA ` ( 1st ` z ) ) )
27	26	oveq2d 6666	. . . . . . . 8 |- ( z e. dom .x. -> ( (domA ` ( 2nd ` z ) ) (Homa ` C ) (coda ` ( 2nd ` z ) ) ) = ( (domA ` ( 2nd ` z ) ) (Homa ` C ) (domA ` ( 1st ` z ) ) ) )
28	25, 27	eleqtrd 2703	. . . . . . 7 |- ( z e. dom .x. -> ( 2nd ` z ) e. ( (domA ` ( 2nd ` z ) ) (Homa ` C ) (domA ` ( 1st ` z ) ) ) )
29	22	simp2d 1074	. . . . . . . 8 |- ( z e. dom .x. -> ( 1st ` z ) e. A )
30	2, 15	arwhoma 16695	. . . . . . . 8 |- ( ( 1st ` z ) e. A -> ( 1st ` z ) e. ( (domA ` ( 1st ` z ) ) (Homa ` C ) (coda ` ( 1st ` z ) ) ) )
31	29, 30	syl 17	. . . . . . 7 |- ( z e. dom .x. -> ( 1st ` z ) e. ( (domA ` ( 1st ` z ) ) (Homa ` C ) (coda ` ( 1st ` z ) ) ) )
32	1, 15, 28, 31	coahom 16720	. . . . . 6 |- ( z e. dom .x. -> ( ( 1st ` z ) .x. ( 2nd ` z ) ) e. ( (domA ` ( 2nd ` z ) ) (Homa ` C ) (coda ` ( 1st ` z ) ) ) )
33	16, 32	sseldi 3601	. . . . 5 |- ( z e. dom .x. -> ( ( 1st ` z ) .x. ( 2nd ` z ) ) e. A )
34	14, 33	eqeltrd 2701	. . . 4 |- ( z e. dom .x. -> ( .x. ` z ) e. A )
35	34	rgen 2922	. . 3 |- A. z e. dom .x. ( .x. ` z ) e. A
36		ffnfv 6388	. . 3 |- ( .x. : dom .x. --> A <-> ( .x. Fn dom .x. /\ A. z e. dom .x. ( .x. ` z ) e. A ) )
37	7, 35, 36	mpbir2an 955	. 2 |- .x. : dom .x. --> A
38		fvex 6201	. . . 4 |- (Nat ` C ) e. _V
39	2, 38	eqeltri 2697	. . 3 |- A e. _V
40	39, 39	xpex 6962	. . 3 |- ( A X. A ) e. _V
41	39, 40	elpm2 7889	. 2 |- ( .x. e. ( A ^pm ( A X. A ) ) <-> ( .x. : dom .x. --> A /\ dom .x. C_ ( A X. A ) ) )
42	37, 8, 41	mpbir2an 955	1 |- .x. e. ( A ^pm ( A X. A ) )

Syntax hints:   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   <.cop 4183   <.cotp 4185   class class class wbr 4653   X. cxp 5112   dom cdm 5114   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   ^pm cpm 7858  compcco 15953  domAcdoma 16670  cod_accoda 16671  Natcarw 16672  Hom_achoma 16673  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-pm 7860  df-cat 16329  df-doma 16674  df-coda 16675  df-homa 16676  df-arw 16677  df-coa 16706

This theorem is referenced by: (None)