Theorem reldmtpos 6999

Description: Necessary and sufficient condition for dom tpos F to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
reldmtpos	|- ( Rel dom tpos F <-> -. (/) e. dom F )

Proof of Theorem reldmtpos

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4528	. . . . 5 |- (/) e. _V
2	1	eldm 5037	. . . 4 |- ( (/) e. dom F <-> E. y (/) F y )
3		vex 3034	. . . . . . 7 |- y e. _V
4		brtpos0 6998	. . . . . . 7 |- ( y e. _V -> ( (/)tpos F y <-> (/) F y ) )
5	3, 4	ax-mp 5	. . . . . 6 |- ( (/)tpos F y <-> (/) F y )
6		0nelxp 4867	. . . . . . . 8 |- -. (/) e. ( _V X. _V )
7		df-rel 4846	. . . . . . . . 9 |- ( Rel dom tpos F <-> dom tpos F C_ ( _V X. _V ) )
8		ssel 3412	. . . . . . . . 9 |- ( dom tpos F C_ ( _V X. _V ) -> ( (/) e. dom tpos F -> (/) e. ( _V X. _V ) ) )
9	7, 8	sylbi 200	. . . . . . . 8 |- ( Rel dom tpos F -> ( (/) e. dom tpos F -> (/) e. ( _V X. _V ) ) )
10	6, 9	mtoi 183	. . . . . . 7 |- ( Rel dom tpos F -> -. (/) e. dom tpos F )
11	1, 3	breldm 5045	. . . . . . 7 |- ( (/)tpos F y -> (/) e. dom tpos F )
12	10, 11	nsyl3 123	. . . . . 6 |- ( (/)tpos F y -> -. Rel dom tpos F )
13	5, 12	sylbir 218	. . . . 5 |- ( (/) F y -> -. Rel dom tpos F )
14	13	exlimiv 1784	. . . 4 |- ( E. y (/) F y -> -. Rel dom tpos F )
15	2, 14	sylbi 200	. . 3 |- ( (/) e. dom F -> -. Rel dom tpos F )
16	15	con2i 124	. 2 |- ( Rel dom tpos F -> -. (/) e. dom F )
17		vex 3034	. . . . . 6 |- x e. _V
18	17	eldm 5037	. . . . 5 |- ( x e. dom tpos F <-> E. y xtpos F y )
19		relcnv 5213	. . . . . . . . . . 11 |- Rel `' dom F
20		df-rel 4846	. . . . . . . . . . 11 |- ( Rel `' dom F <-> `' dom F C_ ( _V X. _V ) )
21	19, 20	mpbi 213	. . . . . . . . . 10 |- `' dom F C_ ( _V X. _V )
22	21	sseli 3414	. . . . . . . . 9 |- ( x e. `' dom F -> x e. ( _V X. _V ) )
23	22	a1i 11	. . . . . . . 8 |- ( ( -. (/) e. dom F /\ xtpos F y ) -> ( x e. `' dom F -> x e. ( _V X. _V ) ) )
24		elsni 3985	. . . . . . . . . . . 12 |- ( x e. { (/) } -> x = (/) )
25	24	breq1d 4405	. . . . . . . . . . 11 |- ( x e. { (/) } -> ( xtpos F y <-> (/)tpos F y ) )
26	1, 3	breldm 5045	. . . . . . . . . . . . 13 |- ( (/) F y -> (/) e. dom F )
27	26	pm2.24d 139	. . . . . . . . . . . 12 |- ( (/) F y -> ( -. (/) e. dom F -> x e. ( _V X. _V ) ) )
28	5, 27	sylbi 200	. . . . . . . . . . 11 |- ( (/)tpos F y -> ( -. (/) e. dom F -> x e. ( _V X. _V ) ) )
29	25, 28	syl6bi 236	. . . . . . . . . 10 |- ( x e. { (/) } -> ( xtpos F y -> ( -. (/) e. dom F -> x e. ( _V X. _V ) ) ) )
30	29	com3l 83	. . . . . . . . 9 |- ( xtpos F y -> ( -. (/) e. dom F -> ( x e. { (/) } -> x e. ( _V X. _V ) ) ) )
31	30	impcom 437	. . . . . . . 8 |- ( ( -. (/) e. dom F /\ xtpos F y ) -> ( x e. { (/) } -> x e. ( _V X. _V ) ) )
32		brtpos2 6997	. . . . . . . . . . . 12 |- ( y e. _V -> ( xtpos F y <-> ( x e. ( `' dom F u. { (/) } ) /\ U. `' { x } F y ) ) )
33	3, 32	ax-mp 5	. . . . . . . . . . 11 |- ( xtpos F y <-> ( x e. ( `' dom F u. { (/) } ) /\ U. `' { x } F y ) )
34	33	simplbi 467	. . . . . . . . . 10 |- ( xtpos F y -> x e. ( `' dom F u. { (/) } ) )
35		elun 3565	. . . . . . . . . 10 |- ( x e. ( `' dom F u. { (/) } ) <-> ( x e. `' dom F \/ x e. { (/) } ) )
36	34, 35	sylib 201	. . . . . . . . 9 |- ( xtpos F y -> ( x e. `' dom F \/ x e. { (/) } ) )
37	36	adantl 473	. . . . . . . 8 |- ( ( -. (/) e. dom F /\ xtpos F y ) -> ( x e. `' dom F \/ x e. { (/) } ) )
38	23, 31, 37	mpjaod 388	. . . . . . 7 |- ( ( -. (/) e. dom F /\ xtpos F y ) -> x e. ( _V X. _V ) )
39	38	ex 441	. . . . . 6 |- ( -. (/) e. dom F -> ( xtpos F y -> x e. ( _V X. _V ) ) )
40	39	exlimdv 1787	. . . . 5 |- ( -. (/) e. dom F -> ( E. y xtpos F y -> x e. ( _V X. _V ) ) )
41	18, 40	syl5bi 225	. . . 4 |- ( -. (/) e. dom F -> ( x e. dom tpos F -> x e. ( _V X. _V ) ) )
42	41	ssrdv 3424	. . 3 |- ( -. (/) e. dom F -> dom tpos F C_ ( _V X. _V ) )
43	42, 7	sylibr 217	. 2 |- ( -. (/) e. dom F -> Rel dom tpos F )
44	16, 43	impbii 192	1 |- ( Rel dom tpos F <-> -. (/) e. dom F )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 375   /\ wa 376   E.wex 1671   e. wcel 1904   _Vcvv 3031   u. cun 3388   C_ wss 3390   (/)c0 3722   {csn 3959   U.cuni 4190   class class class wbr 4395   X. cxp 4837   `'ccnv 4838   dom cdm 4839   Rel wrel 4844  tpos ctpos 6990

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-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-rab 2765  df-v 3033  df-sbc 3256  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-uni 4191  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-fv 5597  df-tpos 6991

This theorem is referenced by:  dmtpos  7003