Theorem rnmpt0 39412

Description: The range of a function in map-to notation is empty if and only if its domain is empty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Hypotheses

Ref	Expression
rnmpt0.1	|- F/ x ph
rnmpt0.2	|- ( ( ph /\ x e. A ) -> B e. V )
rnmpt0.3	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
rnmpt0	|- ( ph -> ( ran F = (/) <-> A = (/) ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   B( x)   F( x)   V( x)

Proof of Theorem rnmpt0

Step	Hyp	Ref	Expression
1		rnmpt0.1	. . . . . 6 |- F/ x ph
2		rnmpt0.2	. . . . . . 7 |- ( ( ph /\ x e. A ) -> B e. V )
3	2	ex 450	. . . . . 6 |- ( ph -> ( x e. A -> B e. V ) )
4	1, 3	ralrimi 2957	. . . . 5 |- ( ph -> A. x e. A B e. V )
5		dmmptg 5632	. . . . 5 |- ( A. x e. A B e. V -> dom ( x e. A |-> B ) = A )
6	4, 5	syl 17	. . . 4 |- ( ph -> dom ( x e. A |-> B ) = A )
7	6	eqcomd 2628	. . 3 |- ( ph -> A = dom ( x e. A |-> B ) )
8	7	eqeq1d 2624	. 2 |- ( ph -> ( A = (/) <-> dom ( x e. A |-> B ) = (/) ) )
9		dm0rn0 5342	. . 3 |- ( dom ( x e. A |-> B ) = (/) <-> ran ( x e. A |-> B ) = (/) )
10	9	a1i 11	. 2 |- ( ph -> ( dom ( x e. A |-> B ) = (/) <-> ran ( x e. A |-> B ) = (/) ) )
11		rnmpt0.3	. . . . . 6 |- F = ( x e. A |-> B )
12	11	rneqi 5352	. . . . 5 |- ran F = ran ( x e. A |-> B )
13	12	a1i 11	. . . 4 |- ( ph -> ran F = ran ( x e. A |-> B ) )
14	13	eqcomd 2628	. . 3 |- ( ph -> ran ( x e. A |-> B ) = ran F )
15	14	eqeq1d 2624	. 2 |- ( ph -> ( ran ( x e. A |-> B ) = (/) <-> ran F = (/) ) )
16	8, 10, 15	3bitrrd 295	1 |- ( ph -> ( ran F = (/) <-> A = (/) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   (/)c0 3915   |-> cmpt 4729   dom cdm 5114   ran crn 5115

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-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-mpt 4730  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  rnmptn0  39413