Theorem 0ima 6043

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 6036	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5881	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3970	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4340	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3938	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1542  ∅c0 4273  ran crn 5632   “ cima 5634

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 2708  ax-sep 5231  ax-pr 5375

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644

This theorem is referenced by:  csbrn  6167  nghmfval  24687  isnghm  24688  mptiffisupp  32766  vieta  33724  mthmval  35757  ec0  38698  0he  44209  limsup0  46122  0cnf  46305