Theorem 0ima 5931

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 5925	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5780	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3923	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4297	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3903	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1543  ∅c0 4223  ran crn 5537   “ cima 5539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-br 5040  df-opab 5102  df-xp 5542  df-cnv 5544  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549

This theorem is referenced by:  csbrn  6046  nghmfval  23574  isnghm  23575  mthmval  33204  ec0  36185  0he  41008  limsup0  42853  0cnf  43036