Theorem rnmptn0 39413

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

Hypotheses

Ref	Expression
rnmptn0.x	|- F/ x ph
rnmptn0.b	|- ( ( ph /\ x e. A ) -> B e. V )
rnmptn0.f	|- F = ( x e. A |-> B )
rnmptn0.a	|- ( ph -> A =/= (/) )

Assertion

Ref	Expression
rnmptn0	|- ( ph -> ran F =/= (/) )

Distinct variable group:   x, A

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

Proof of Theorem rnmptn0

Step	Hyp	Ref	Expression
1		rnmptn0.a	. . . 4 |- ( ph -> A =/= (/) )
2	1	neneqd 2799	. . 3 |- ( ph -> -. A = (/) )
3		rnmptn0.x	. . . 4 |- F/ x ph
4		rnmptn0.b	. . . 4 |- ( ( ph /\ x e. A ) -> B e. V )
5		rnmptn0.f	. . . 4 |- F = ( x e. A |-> B )
6	3, 4, 5	rnmpt0 39412	. . 3 |- ( ph -> ( ran F = (/) <-> A = (/) ) )
7	2, 6	mtbird 315	. 2 |- ( ph -> -. ran F = (/) )
8	7	neqned 2801	1 |- ( ph -> ran F =/= (/) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   =/= wne 2794   (/)c0 3915   |-> cmpt 4729   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-ne 2795  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:  infnsuprnmpt  39465  suprclrnmpt  39466  fisupclrnmpt  39622  supxrrernmpt  39648  suprleubrnmpt  39649  supxrre3rnmpt  39656  supminfrnmpt  39672  infrpgernmpt  39695  limsupvaluz2  39970  ioorrnopnlem  40524  iunhoiioolem  40889  vonioolem1  40894