Theorem 0ima 6213

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 6210	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5915	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3961	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4353	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3929	1 ⊢ (∅ “ 𝐴) = ∅

This proof depends on syntax axioms:   = wceq 1570  ∅c0 4281  ran crn 5656   “ cima 5658

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668

This theorem is used by:  csbrn  6217  nghmfval  25071  isnghm  25072  mptiffisupp  33357  vieta  34282  mthmval  36507  ec0  39501  0he  44974  limsup0  46909  0cnf  47092