Theorem 0ima 6037

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 6030	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5875	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3982	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4352	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3950	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1541  ∅c0 4285  ran crn 5625   “ cima 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637

This theorem is referenced by:  csbrn  6161  nghmfval  24666  isnghm  24667  mptiffisupp  32772  vieta  33736  mthmval  35769  ec0  38558  0he  44019  limsup0  45934  0cnf  46117