Theorem 0ima 6027

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 6020	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5866	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3983	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4350	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3951	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1541  ∅c0 4283  ran crn 5617   “ cima 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629

This theorem is referenced by:  csbrn  6150  nghmfval  24635  isnghm  24636  mptiffisupp  32669  mthmval  35607  ec0  38396  0he  43814  limsup0  45731  0cnf  45914