Theorem reldmtpos 6955

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 4569	. . . . 5 |- (/) e. _V
2	1	eldm 5189	. . . 4 |- ( (/) e. dom F <-> E. y (/) F y )
3		vex 3109	. . . . . . 7 |- y e. _V
4		brtpos0 6954	. . . . . . 7 |- ( y e. _V -> ( (/)tpos F y <-> (/) F y ) )
5	3, 4	ax-mp 5	. . . . . 6 |- ( (/)tpos F y <-> (/) F y )
6		0nelxp 5016	. . . . . . . 8 |- -. (/) e. ( _V X. _V )
7		df-rel 4995	. . . . . . . . 9 |- ( Rel dom tpos F <-> dom tpos F C_ ( _V X. _V ) )
8		ssel 3483	. . . . . . . . 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 5196	. . . . . . 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 1727	. . . 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 3109	. . . . . 6 |- x e. _V
18	17	eldm 5189	. . . . 5 |- ( x e. dom tpos F <-> E. y xtpos F y )
19		relcnv 5362	. . . . . . . . . . 11 |- Rel `' dom F
20		df-rel 4995	. . . . . . . . . . 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 3485	. . . . . . . . 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 4041	. . . . . . . . . . . 12 |- ( x e. { (/) } -> x = (/) )
25	24	breq1d 4449	. . . . . . . . . . 11 |- ( x e. { (/) } -> ( xtpos F y <-> (/)tpos F y ) )
26	1, 3	breldm 5196	. . . . . . . . . . . . 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 428	. . . . . . . 8 |- ( ( -. (/) e. dom F /\ xtpos F y ) -> ( x e. { (/) } -> x e. ( _V X. _V ) ) )
32		brtpos2 6953	. . . . . . . . . . . 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 458	. . . . . . . . . 10 |- ( xtpos F y -> x e. ( `' dom F u. { (/) } ) )
35		elun 3631	. . . . . . . . . 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 464	. . . . . . . 8 |- ( ( -. (/) e. dom F /\ xtpos F y ) -> ( x e. `' dom F \/ x e. { (/) } ) )
38	23, 31, 37	mpjaod 379	. . . . . . 7 |- ( ( -. (/) e. dom F /\ xtpos F y ) -> x e. ( _V X. _V ) )
39	38	ex 432	. . . . . 6 |- ( -. (/) e. dom F -> ( xtpos F y -> x e. ( _V X. _V ) ) )
40	39	exlimdv 1729	. . . . 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 3495	. . 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 366   /\ wa 367   E.wex 1617   e. wcel 1823   _Vcvv 3106   u. cun 3459   C_ wss 3461   (/)c0 3783   {csn 4016   U.cuni 4235   class class class wbr 4439   X. cxp 4986   `'ccnv 4987   dom cdm 4988   Rel wrel 4993  tpos ctpos 6946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578  df-tpos 6947

This theorem is referenced by:  dmtpos  6959