Theorem f0rn0 5392

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 5353	. . 3 |- ( E : X --> Y -> dom E = X )
2		frn 5356	. . . . . . . . 9 |- ( E : X --> Y -> ran E C_ Y )
3		ralnex 2458	. . . . . . . . . 10 |- ( A. y e. Y -. y e. ran E <-> -. E. y e. Y y e. ran E )
4		disj 3463	. . . . . . . . . . 11 |- ( ( Y i^i ran E ) = (/) <-> A. y e. Y -. y e. ran E )
5		df-ss 3134	. . . . . . . . . . . 12 |- ( ran E C_ Y <-> ( ran E i^i Y ) = ran E )
6		incom 3319	. . . . . . . . . . . . . 14 |- ( ran E i^i Y ) = ( Y i^i ran E )
7	6	eqeq1i 2178	. . . . . . . . . . . . 13 |- ( ( ran E i^i Y ) = ran E <-> ( Y i^i ran E ) = ran E )
8		eqtr2 2189	. . . . . . . . . . . . . 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 4828	. . . . . 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 2177	. . . . . . 7 |- ( X = dom E -> ( X = (/) <-> dom E = (/) ) )
20	19	eqcoms 2173	. . . . . 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 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   i^i cin 3120   C_ wss 3121   (/)c0 3414   dom cdm 4611   ran crn 4612   -->wf 5194

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-cnv 4619  df-dm 4621  df-rn 4622  df-fn 5201  df-f 5202

This theorem is referenced by: (None)