Theorem f0rn0 6090

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 6051	. . 3 |- ( E : X --> Y -> dom E = X )
2		frn 6053	. . . . . . . . 9 |- ( E : X --> Y -> ran E C_ Y )
3		ralnex 2992	. . . . . . . . . 10 |- ( A. y e. Y -. y e. ran E <-> -. E. y e. Y y e. ran E )
4		disj 4017	. . . . . . . . . . 11 |- ( ( Y i^i ran E ) = (/) <-> A. y e. Y -. y e. ran E )
5		df-ss 3588	. . . . . . . . . . . 12 |- ( ran E C_ Y <-> ( ran E i^i Y ) = ran E )
6		incom 3805	. . . . . . . . . . . . . 14 |- ( ran E i^i Y ) = ( Y i^i ran E )
7	6	eqeq1i 2627	. . . . . . . . . . . . 13 |- ( ( ran E i^i Y ) = ran E <-> ( Y i^i ran E ) = ran E )
8		eqtr2 2642	. . . . . . . . . . . . . 14 |- ( ( ( Y i^i ran E ) = ran E /\ ( Y i^i ran E ) = (/) ) -> ran E = (/) )
9	8	ex 450	. . . . . . . . . . . . 13 |- ( ( Y i^i ran E ) = ran E -> ( ( Y i^i ran E ) = (/) -> ran E = (/) ) )
10	7, 9	sylbi 207	. . . . . . . . . . . 12 |- ( ( ran E i^i Y ) = ran E -> ( ( Y i^i ran E ) = (/) -> ran E = (/) ) )
11	5, 10	sylbi 207	. . . . . . . . . . 11 |- ( ran E C_ Y -> ( ( Y i^i ran E ) = (/) -> ran E = (/) ) )
12	4, 11	syl5bir 233	. . . . . . . . . 10 |- ( ran E C_ Y -> ( A. y e. Y -. y e. ran E -> ran E = (/) ) )
13	3, 12	syl5bir 233	. . . . . . . . 9 |- ( ran E C_ Y -> ( -. E. y e. Y y e. ran E -> ran E = (/) ) )
14	2, 13	syl 17	. . . . . . . 8 |- ( E : X --> Y -> ( -. E. y e. Y y e. ran E -> ran E = (/) ) )
15	14	imp 445	. . . . . . 7 |- ( ( E : X --> Y /\ -. E. y e. Y y e. ran E ) -> ran E = (/) )
16	15	adantl 482	. . . . . 6 |- ( ( dom E = X /\ ( E : X --> Y /\ -. E. y e. Y y e. ran E ) ) -> ran E = (/) )
17		dm0rn0 5342	. . . . . 6 |- ( dom E = (/) <-> ran E = (/) )
18	16, 17	sylibr 224	. . . . 5 |- ( ( dom E = X /\ ( E : X --> Y /\ -. E. y e. Y y e. ran E ) ) -> dom E = (/) )
19		eqeq1 2626	. . . . . . 7 |- ( X = dom E -> ( X = (/) <-> dom E = (/) ) )
20	19	eqcoms 2630	. . . . . 6 |- ( dom E = X -> ( X = (/) <-> dom E = (/) ) )
21	20	adantr 481	. . . . 5 |- ( ( dom E = X /\ ( E : X --> Y /\ -. E. y e. Y y e. ran E ) ) -> ( X = (/) <-> dom E = (/) ) )
22	18, 21	mpbird 247	. . . 4 |- ( ( dom E = X /\ ( E : X --> Y /\ -. E. y e. Y y e. ran E ) ) -> X = (/) )
23	22	exp32 631	. . 3 |- ( dom E = X -> ( E : X --> Y -> ( -. E. y e. Y y e. ran E -> X = (/) ) ) )
24	1, 23	mpcom 38	. 2 |- ( E : X --> Y -> ( -. E. y e. Y y e. ran E -> X = (/) ) )
25	24	imp 445	1 |- ( ( E : X --> Y /\ -. E. y e. Y y e. ran E ) -> X = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   (/)c0 3915   dom cdm 5114   ran crn 5115   -->wf 5884

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-fn 5891  df-f 5892

This theorem is referenced by: (None)