Theorem 0ima 6045

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 6038	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5883	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3984	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4354	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3952	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1542  ∅c0 4287  ran crn 5633   “ cima 5635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645

This theorem is referenced by:  csbrn  6169  nghmfval  24678  isnghm  24679  mptiffisupp  32782  vieta  33756  mthmval  35788  ec0  38622  0he  44132  limsup0  46046  0cnf  46229