Theorem 0ima 6078

Description: Image under the empty relation. (Contributed by FL, 11-Jan-2007.)

Assertion

Ref	Expression
0ima	⊢ (∅ “ 𝐴) = ∅

Proof of Theorem 0ima

Step	Hyp	Ref	Expression
1		imassrn 6071	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5914	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3982	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4353	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3950	1 ⊢ (∅ “ 𝐴) = ∅

This proof depends on syntax axioms:   = wceq 1570  ∅c0 4282  ran crn 5660   “ cima 5662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672

This theorem is used by:  csbrn  6203  nghmfval  24954  isnghm  24955  mptiffisupp  33173  vieta  34098  mthmval  36162  ec0  39133  0he  44630  limsup0  46530  0cnf  46713