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 3971	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4341	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3939	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1542  ∅c0 4274  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 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  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  24697  isnghm  24698  mptiffisupp  32781  vieta  33739  mthmval  35773  ec0  38712  0he  44227  limsup0  46140  0cnf  46323