Theorem reldmtpos 7360

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 4790	. . . . 5 |- (/) e. _V
2	1	eldm 5321	. . . 4 |- ( (/) e. dom F <-> E. y (/) F y )
3		vex 3203	. . . . . . 7 |- y e. _V
4		brtpos0 7359	. . . . . . 7 |- ( y e. _V -> ( (/)tpos F y <-> (/) F y ) )
5	3, 4	ax-mp 5	. . . . . 6 |- ( (/)tpos F y <-> (/) F y )
6		0nelxp 5143	. . . . . . . 8 |- -. (/) e. ( _V X. _V )
7		df-rel 5121	. . . . . . . . 9 |- ( Rel dom tpos F <-> dom tpos F C_ ( _V X. _V ) )
8		ssel 3597	. . . . . . . . 9 |- ( dom tpos F C_ ( _V X. _V ) -> ( (/) e. dom tpos F -> (/) e. ( _V X. _V ) ) )
9	7, 8	sylbi 207	. . . . . . . 8 |- ( Rel dom tpos F -> ( (/) e. dom tpos F -> (/) e. ( _V X. _V ) ) )
10	6, 9	mtoi 190	. . . . . . 7 |- ( Rel dom tpos F -> -. (/) e. dom tpos F )
11	1, 3	breldm 5329	. . . . . . 7 |- ( (/)tpos F y -> (/) e. dom tpos F )
12	10, 11	nsyl3 133	. . . . . 6 |- ( (/)tpos F y -> -. Rel dom tpos F )
13	5, 12	sylbir 225	. . . . 5 |- ( (/) F y -> -. Rel dom tpos F )
14	13	exlimiv 1858	. . . 4 |- ( E. y (/) F y -> -. Rel dom tpos F )
15	2, 14	sylbi 207	. . 3 |- ( (/) e. dom F -> -. Rel dom tpos F )
16	15	con2i 134	. 2 |- ( Rel dom tpos F -> -. (/) e. dom F )
17		vex 3203	. . . . . 6 |- x e. _V
18	17	eldm 5321	. . . . 5 |- ( x e. dom tpos F <-> E. y xtpos F y )
19		relcnv 5503	. . . . . . . . . . 11 |- Rel `' dom F
20		df-rel 5121	. . . . . . . . . . 11 |- ( Rel `' dom F <-> `' dom F C_ ( _V X. _V ) )
21	19, 20	mpbi 220	. . . . . . . . . 10 |- `' dom F C_ ( _V X. _V )
22	21	sseli 3599	. . . . . . . . 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 4194	. . . . . . . . . . . 12 |- ( x e. { (/) } -> x = (/) )
25	24	breq1d 4663	. . . . . . . . . . 11 |- ( x e. { (/) } -> ( xtpos F y <-> (/)tpos F y ) )
26	1, 3	breldm 5329	. . . . . . . . . . . . 13 |- ( (/) F y -> (/) e. dom F )
27	26	pm2.24d 147	. . . . . . . . . . . 12 |- ( (/) F y -> ( -. (/) e. dom F -> x e. ( _V X. _V ) ) )
28	5, 27	sylbi 207	. . . . . . . . . . 11 |- ( (/)tpos F y -> ( -. (/) e. dom F -> x e. ( _V X. _V ) ) )
29	25, 28	syl6bi 243	. . . . . . . . . 10 |- ( x e. { (/) } -> ( xtpos F y -> ( -. (/) e. dom F -> x e. ( _V X. _V ) ) ) )
30	29	com3l 89	. . . . . . . . 9 |- ( xtpos F y -> ( -. (/) e. dom F -> ( x e. { (/) } -> x e. ( _V X. _V ) ) ) )
31	30	impcom 446	. . . . . . . 8 |- ( ( -. (/) e. dom F /\ xtpos F y ) -> ( x e. { (/) } -> x e. ( _V X. _V ) ) )
32		brtpos2 7358	. . . . . . . . . . . 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 476	. . . . . . . . . 10 |- ( xtpos F y -> x e. ( `' dom F u. { (/) } ) )
35		elun 3753	. . . . . . . . . 10 |- ( x e. ( `' dom F u. { (/) } ) <-> ( x e. `' dom F \/ x e. { (/) } ) )
36	34, 35	sylib 208	. . . . . . . . 9 |- ( xtpos F y -> ( x e. `' dom F \/ x e. { (/) } ) )
37	36	adantl 482	. . . . . . . 8 |- ( ( -. (/) e. dom F /\ xtpos F y ) -> ( x e. `' dom F \/ x e. { (/) } ) )
38	23, 31, 37	mpjaod 396	. . . . . . 7 |- ( ( -. (/) e. dom F /\ xtpos F y ) -> x e. ( _V X. _V ) )
39	38	ex 450	. . . . . 6 |- ( -. (/) e. dom F -> ( xtpos F y -> x e. ( _V X. _V ) ) )
40	39	exlimdv 1861	. . . . 5 |- ( -. (/) e. dom F -> ( E. y xtpos F y -> x e. ( _V X. _V ) ) )
41	18, 40	syl5bi 232	. . . 4 |- ( -. (/) e. dom F -> ( x e. dom tpos F -> x e. ( _V X. _V ) ) )
42	41	ssrdv 3609	. . 3 |- ( -. (/) e. dom F -> dom tpos F C_ ( _V X. _V ) )
43	42, 7	sylibr 224	. 2 |- ( -. (/) e. dom F -> Rel dom tpos F )
44	16, 43	impbii 199	1 |- ( Rel dom tpos F <-> -. (/) e. dom F )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   E.wex 1704   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   U.cuni 4436   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   dom cdm 5114   Rel wrel 5119  tpos ctpos 7351

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-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-rab 2921  df-v 3202  df-sbc 3436  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-uni 4437  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-fv 5896  df-tpos 7352

This theorem is referenced by:  dmtpos  7364