Theorem f0rn0 5519

Description: If there is no element in the range of a function, its domain must be empty. (Contributed by Alexander van der Vekens, 12-Jul-2018.)

Assertion

Ref	Expression
f0rn0	|- ( ( E : X --> Y /\ -. E. y e. Y y e. ran E ) -> X = (/) )

Distinct variable groups:   y, E   y, Y

Allowed substitution hint:   X( y)

Proof of Theorem f0rn0

Step	Hyp	Ref	Expression
1		fdm 5478	. . 3 |- ( E : X --> Y -> dom E = X )
2		frn 5481	. . . . . . . . 9 |- ( E : X --> Y -> ran E C_ Y )
3		ralnex 2518	. . . . . . . . . 10 |- ( A. y e. Y -. y e. ran E <-> -. E. y e. Y y e. ran E )
4		disj 3540	. . . . . . . . . . 11 |- ( ( Y i^i ran E ) = (/) <-> A. y e. Y -. y e. ran E )
5		df-ss 3210	. . . . . . . . . . . 12 |- ( ran E C_ Y <-> ( ran E i^i Y ) = ran E )
6		incom 3396	. . . . . . . . . . . . . 14 |- ( ran E i^i Y ) = ( Y i^i ran E )
7	6	eqeq1i 2237	. . . . . . . . . . . . 13 |- ( ( ran E i^i Y ) = ran E <-> ( Y i^i ran E ) = ran E )
8		eqtr2 2248	. . . . . . . . . . . . . 14 |- ( ( ( Y i^i ran E ) = ran E /\ ( Y i^i ran E ) = (/) ) -> ran E = (/) )
9	8	ex 115	. . . . . . . . . . . . 13 |- ( ( Y i^i ran E ) = ran E -> ( ( Y i^i ran E ) = (/) -> ran E = (/) ) )
10	7, 9	sylbi 121	. . . . . . . . . . . 12 |- ( ( ran E i^i Y ) = ran E -> ( ( Y i^i ran E ) = (/) -> ran E = (/) ) )
11	5, 10	sylbi 121	. . . . . . . . . . 11 |- ( ran E C_ Y -> ( ( Y i^i ran E ) = (/) -> ran E = (/) ) )
12	4, 11	biimtrrid 153	. . . . . . . . . 10 |- ( ran E C_ Y -> ( A. y e. Y -. y e. ran E -> ran E = (/) ) )
13	3, 12	biimtrrid 153	. . . . . . . . 9 |- ( ran E C_ Y -> ( -. E. y e. Y y e. ran E -> ran E = (/) ) )
14	2, 13	syl 14	. . . . . . . 8 |- ( E : X --> Y -> ( -. E. y e. Y y e. ran E -> ran E = (/) ) )
15	14	imp 124	. . . . . . 7 |- ( ( E : X --> Y /\ -. E. y e. Y y e. ran E ) -> ran E = (/) )
16	15	adantl 277	. . . . . 6 |- ( ( dom E = X /\ ( E : X --> Y /\ -. E. y e. Y y e. ran E ) ) -> ran E = (/) )
17		dm0rn0 4939	. . . . . 6 |- ( dom E = (/) <-> ran E = (/) )
18	16, 17	sylibr 134	. . . . 5 |- ( ( dom E = X /\ ( E : X --> Y /\ -. E. y e. Y y e. ran E ) ) -> dom E = (/) )
19		eqeq1 2236	. . . . . . 7 |- ( X = dom E -> ( X = (/) <-> dom E = (/) ) )
20	19	eqcoms 2232	. . . . . 6 |- ( dom E = X -> ( X = (/) <-> dom E = (/) ) )
21	20	adantr 276	. . . . 5 |- ( ( dom E = X /\ ( E : X --> Y /\ -. E. y e. Y y e. ran E ) ) -> ( X = (/) <-> dom E = (/) ) )
22	18, 21	mpbird 167	. . . 4 |- ( ( dom E = X /\ ( E : X --> Y /\ -. E. y e. Y y e. ran E ) ) -> X = (/) )
23	22	exp32 365	. . 3 |- ( dom E = X -> ( E : X --> Y -> ( -. E. y e. Y y e. ran E -> X = (/) ) ) )
24	1, 23	mpcom 36	. 2 |- ( E : X --> Y -> ( -. E. y e. Y y e. ran E -> X = (/) ) )
25	24	imp 124	1 |- ( ( E : X --> Y /\ -. E. y e. Y y e. ran E ) -> X = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   i^i cin 3196   C_ wss 3197   (/)c0 3491   dom cdm 4718   ran crn 4719   -->wf 5313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-cnv 4726  df-dm 4728  df-rn 4729  df-fn 5320  df-f 5321

This theorem is referenced by: (None)