Theorem 0ima 6098

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 6091	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5939	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 4032	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4406	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 4012	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1537  ∅c0 4339  ran crn 5690   “ cima 5692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702

This theorem is referenced by:  csbrn  6225  nghmfval  24759  isnghm  24760  mptiffisupp  32708  mthmval  35560  ec0  38351  0he  43772  limsup0  45650  0cnf  45833