Theorem reldmtpos 6960

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 4577	. . . . 5 |- (/) e. _V
2	1	eldm 5198	. . . 4 |- ( (/) e. dom F <-> E. y (/) F y )
3		vex 3116	. . . . . . 7 |- y e. _V
4		brtpos0 6959	. . . . . . 7 |- ( y e. _V -> ( (/)tpos F y <-> (/) F y ) )
5	3, 4	ax-mp 5	. . . . . 6 |- ( (/)tpos F y <-> (/) F y )
6		0nelxp 5026	. . . . . . . 8 |- -. (/) e. ( _V X. _V )
7		df-rel 5006	. . . . . . . . 9 |- ( Rel dom tpos F <-> dom tpos F C_ ( _V X. _V ) )
8		ssel 3498	. . . . . . . . 9 |- ( dom tpos F C_ ( _V X. _V ) -> ( (/) e. dom tpos F -> (/) e. ( _V X. _V ) ) )
9	7, 8	sylbi 195	. . . . . . . 8 |- ( Rel dom tpos F -> ( (/) e. dom tpos F -> (/) e. ( _V X. _V ) ) )
10	6, 9	mtoi 178	. . . . . . 7 |- ( Rel dom tpos F -> -. (/) e. dom tpos F )
11	1, 3	breldm 5205	. . . . . . 7 |- ( (/)tpos F y -> (/) e. dom tpos F )
12	10, 11	nsyl3 119	. . . . . 6 |- ( (/)tpos F y -> -. Rel dom tpos F )
13	5, 12	sylbir 213	. . . . 5 |- ( (/) F y -> -. Rel dom tpos F )
14	13	exlimiv 1698	. . . 4 |- ( E. y (/) F y -> -. Rel dom tpos F )
15	2, 14	sylbi 195	. . 3 |- ( (/) e. dom F -> -. Rel dom tpos F )
16	15	con2i 120	. 2 |- ( Rel dom tpos F -> -. (/) e. dom F )
17		vex 3116	. . . . . 6 |- x e. _V
18	17	eldm 5198	. . . . 5 |- ( x e. dom tpos F <-> E. y xtpos F y )
19		relcnv 5372	. . . . . . . . . . 11 |- Rel `' dom F
20		df-rel 5006	. . . . . . . . . . 11 |- ( Rel `' dom F <-> `' dom F C_ ( _V X. _V ) )
21	19, 20	mpbi 208	. . . . . . . . . 10 |- `' dom F C_ ( _V X. _V )
22	21	sseli 3500	. . . . . . . . 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 4052	. . . . . . . . . . . 12 |- ( x e. { (/) } -> x = (/) )
25	24	breq1d 4457	. . . . . . . . . . 11 |- ( x e. { (/) } -> ( xtpos F y <-> (/)tpos F y ) )
26	1, 3	breldm 5205	. . . . . . . . . . . . 13 |- ( (/) F y -> (/) e. dom F )
27	26	pm2.24d 143	. . . . . . . . . . . 12 |- ( (/) F y -> ( -. (/) e. dom F -> x e. ( _V X. _V ) ) )
28	5, 27	sylbi 195	. . . . . . . . . . 11 |- ( (/)tpos F y -> ( -. (/) e. dom F -> x e. ( _V X. _V ) ) )
29	25, 28	syl6bi 228	. . . . . . . . . 10 |- ( x e. { (/) } -> ( xtpos F y -> ( -. (/) e. dom F -> x e. ( _V X. _V ) ) ) )
30	29	com3l 81	. . . . . . . . 9 |- ( xtpos F y -> ( -. (/) e. dom F -> ( x e. { (/) } -> x e. ( _V X. _V ) ) ) )
31	30	impcom 430	. . . . . . . 8 |- ( ( -. (/) e. dom F /\ xtpos F y ) -> ( x e. { (/) } -> x e. ( _V X. _V ) ) )
32		brtpos2 6958	. . . . . . . . . . . 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 460	. . . . . . . . . 10 |- ( xtpos F y -> x e. ( `' dom F u. { (/) } ) )
35		elun 3645	. . . . . . . . . 10 |- ( x e. ( `' dom F u. { (/) } ) <-> ( x e. `' dom F \/ x e. { (/) } ) )
36	34, 35	sylib 196	. . . . . . . . 9 |- ( xtpos F y -> ( x e. `' dom F \/ x e. { (/) } ) )
37	36	adantl 466	. . . . . . . 8 |- ( ( -. (/) e. dom F /\ xtpos F y ) -> ( x e. `' dom F \/ x e. { (/) } ) )
38	23, 31, 37	mpjaod 381	. . . . . . 7 |- ( ( -. (/) e. dom F /\ xtpos F y ) -> x e. ( _V X. _V ) )
39	38	ex 434	. . . . . 6 |- ( -. (/) e. dom F -> ( xtpos F y -> x e. ( _V X. _V ) ) )
40	39	exlimdv 1700	. . . . 5 |- ( -. (/) e. dom F -> ( E. y xtpos F y -> x e. ( _V X. _V ) ) )
41	18, 40	syl5bi 217	. . . 4 |- ( -. (/) e. dom F -> ( x e. dom tpos F -> x e. ( _V X. _V ) ) )
42	41	ssrdv 3510	. . 3 |- ( -. (/) e. dom F -> dom tpos F C_ ( _V X. _V ) )
43	42, 7	sylibr 212	. 2 |- ( -. (/) e. dom F -> Rel dom tpos F )
44	16, 43	impbii 188	1 |- ( Rel dom tpos F <-> -. (/) e. dom F )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   E.wex 1596   e. wcel 1767   _Vcvv 3113   u. cun 3474   C_ wss 3476   (/)c0 3785   {csn 4027   U.cuni 4245   class class class wbr 4447   X. cxp 4997   `'ccnv 4998   dom cdm 4999   Rel wrel 5004  tpos ctpos 6951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-tpos 6952

This theorem is referenced by:  dmtpos  6964