Theorem f0rn0 5382

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 5343	. . 3 |- ( E : X --> Y -> dom E = X )
2		frn 5346	. . . . . . . . 9 |- ( E : X --> Y -> ran E C_ Y )
3		ralnex 2454	. . . . . . . . . 10 |- ( A. y e. Y -. y e. ran E <-> -. E. y e. Y y e. ran E )
4		disj 3457	. . . . . . . . . . 11 |- ( ( Y i^i ran E ) = (/) <-> A. y e. Y -. y e. ran E )
5		df-ss 3129	. . . . . . . . . . . 12 |- ( ran E C_ Y <-> ( ran E i^i Y ) = ran E )
6		incom 3314	. . . . . . . . . . . . . 14 |- ( ran E i^i Y ) = ( Y i^i ran E )
7	6	eqeq1i 2173	. . . . . . . . . . . . 13 |- ( ( ran E i^i Y ) = ran E <-> ( Y i^i ran E ) = ran E )
8		eqtr2 2184	. . . . . . . . . . . . . 14 |- ( ( ( Y i^i ran E ) = ran E /\ ( Y i^i ran E ) = (/) ) -> ran E = (/) )
9	8	ex 114	. . . . . . . . . . . . 13 |- ( ( Y i^i ran E ) = ran E -> ( ( Y i^i ran E ) = (/) -> ran E = (/) ) )
10	7, 9	sylbi 120	. . . . . . . . . . . 12 |- ( ( ran E i^i Y ) = ran E -> ( ( Y i^i ran E ) = (/) -> ran E = (/) ) )
11	5, 10	sylbi 120	. . . . . . . . . . 11 |- ( ran E C_ Y -> ( ( Y i^i ran E ) = (/) -> ran E = (/) ) )
12	4, 11	syl5bir 152	. . . . . . . . . 10 |- ( ran E C_ Y -> ( A. y e. Y -. y e. ran E -> ran E = (/) ) )
13	3, 12	syl5bir 152	. . . . . . . . 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 123	. . . . . . 7 |- ( ( E : X --> Y /\ -. E. y e. Y y e. ran E ) -> ran E = (/) )
16	15	adantl 275	. . . . . 6 |- ( ( dom E = X /\ ( E : X --> Y /\ -. E. y e. Y y e. ran E ) ) -> ran E = (/) )
17		dm0rn0 4821	. . . . . 6 |- ( dom E = (/) <-> ran E = (/) )
18	16, 17	sylibr 133	. . . . 5 |- ( ( dom E = X /\ ( E : X --> Y /\ -. E. y e. Y y e. ran E ) ) -> dom E = (/) )
19		eqeq1 2172	. . . . . . 7 |- ( X = dom E -> ( X = (/) <-> dom E = (/) ) )
20	19	eqcoms 2168	. . . . . 6 |- ( dom E = X -> ( X = (/) <-> dom E = (/) ) )
21	20	adantr 274	. . . . 5 |- ( ( dom E = X /\ ( E : X --> Y /\ -. E. y e. Y y e. ran E ) ) -> ( X = (/) <-> dom E = (/) ) )
22	18, 21	mpbird 166	. . . 4 |- ( ( dom E = X /\ ( E : X --> Y /\ -. E. y e. Y y e. ran E ) ) -> X = (/) )
23	22	exp32 363	. . 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 123	1 |- ( ( E : X --> Y /\ -. E. y e. Y y e. ran E ) -> X = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   i^i cin 3115   C_ wss 3116   (/)c0 3409   dom cdm 4604   ran crn 4605   -->wf 5184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-cnv 4612  df-dm 4614  df-rn 4615  df-fn 5191  df-f 5192

This theorem is referenced by: (None)