Theorem eldmcoa 16715

Description: A pair <. G , F >. is in the domain of the arrow composition, if the domain of G equals the codomain of F. (In this case we say G and F are composable.) (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

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

Assertion

Ref	Expression
eldmcoa	|- ( G dom .x. F <-> ( F e. A /\ G e. A /\ (coda ` F ) = (domA ` G ) ) )

Proof of Theorem eldmcoa

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

Step	Hyp	Ref	Expression
1		df-br 4654	. 2 |- ( G dom .x. F <-> <. G , F >. e. dom .x. )
2		otex 4933	. . . . . 6 |- <. (domA ` f ) , (coda ` g ) , ( ( 2nd ` g ) ( <. (domA ` f ) , (domA ` g ) >. (comp ` C ) (coda ` g ) ) ( 2nd ` f ) ) >. e. _V
3	2	rgen2w 2925	. . . . 5 |- A. g e. A 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 ) ) >. e. _V
4		coafval.o	. . . . . . 7 |- .x. = (compa ` C )
5		coafval.a	. . . . . . 7 |- A = (Nat ` C )
6		eqid 2622	. . . . . . 7 |- (comp ` C ) = (comp ` C )
7	4, 5, 6	coafval 16714	. . . . . 6 |- .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 ) ) >. )
8	7	fmpt2x 7236	. . . . 5 |- ( A. g e. A 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 ) ) >. e. _V <-> .x. : U_ g e. A ( { g } X. { h e. A | (coda ` h ) = (domA ` g ) } ) --> _V )
9	3, 8	mpbi 220	. . . 4 |- .x. : U_ g e. A ( { g } X. { h e. A | (coda ` h ) = (domA ` g ) } ) --> _V
10	9	fdmi 6052	. . 3 |- dom .x. = U_ g e. A ( { g } X. { h e. A | (coda ` h ) = (domA ` g ) } )
11	10	eleq2i 2693	. 2 |- ( <. G , F >. e. dom .x. <-> <. G , F >. e. U_ g e. A ( { g } X. { h e. A | (coda ` h ) = (domA ` g ) } ) )
12		fveq2 6191	. . . . . 6 |- ( g = G -> (domA ` g ) = (domA ` G ) )
13	12	eqeq2d 2632	. . . . 5 |- ( g = G -> ( (coda ` h ) = (domA ` g ) <-> (coda ` h ) = (domA ` G ) ) )
14	13	rabbidv 3189	. . . 4 |- ( g = G -> { h e. A | (coda ` h ) = (domA ` g ) } = { h e. A | (coda ` h ) = (domA ` G ) } )
15	14	opeliunxp2 5260	. . 3 |- ( <. G , F >. e. U_ g e. A ( { g } X. { h e. A | (coda ` h ) = (domA ` g ) } ) <-> ( G e. A /\ F e. { h e. A | (coda ` h ) = (domA ` G ) } ) )
16		fveq2 6191	. . . . . 6 |- ( h = F -> (coda ` h ) = (coda ` F ) )
17	16	eqeq1d 2624	. . . . 5 |- ( h = F -> ( (coda ` h ) = (domA ` G ) <-> (coda ` F ) = (domA ` G ) ) )
18	17	elrab 3363	. . . 4 |- ( F e. { h e. A | (coda ` h ) = (domA ` G ) } <-> ( F e. A /\ (coda ` F ) = (domA ` G ) ) )
19	18	anbi2i 730	. . 3 |- ( ( G e. A /\ F e. { h e. A | (coda ` h ) = (domA ` G ) } ) <-> ( G e. A /\ ( F e. A /\ (coda ` F ) = (domA ` G ) ) ) )
20		an12 838	. . . 4 |- ( ( G e. A /\ ( F e. A /\ (coda ` F ) = (domA ` G ) ) ) <-> ( F e. A /\ ( G e. A /\ (coda ` F ) = (domA ` G ) ) ) )
21		3anass 1042	. . . 4 |- ( ( F e. A /\ G e. A /\ (coda ` F ) = (domA ` G ) ) <-> ( F e. A /\ ( G e. A /\ (coda ` F ) = (domA ` G ) ) ) )
22	20, 21	bitr4i 267	. . 3 |- ( ( G e. A /\ ( F e. A /\ (coda ` F ) = (domA ` G ) ) ) <-> ( F e. A /\ G e. A /\ (coda ` F ) = (domA ` G ) ) )
23	15, 19, 22	3bitri 286	. 2 |- ( <. G , F >. e. U_ g e. A ( { g } X. { h e. A | (coda ` h ) = (domA ` g ) } ) <-> ( F e. A /\ G e. A /\ (coda ` F ) = (domA ` G ) ) )
24	1, 11, 23	3bitri 286	1 |- ( G dom .x. F <-> ( F e. A /\ G e. A /\ (coda ` F ) = (domA ` G ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   {csn 4177   <.cop 4183   <.cotp 4185   U_ciun 4520   class class class wbr 4653   X. cxp 5112   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   2ndc2nd 7167  compcco 15953  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:  homdmcoa  16717  coapm  16721