Theorem 0ima 6096

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 6089	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5936	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 4032	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4400	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 4000	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1540  ∅c0 4333  ran crn 5686   “ cima 5688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698

This theorem is referenced by:  csbrn  6223  nghmfval  24743  isnghm  24744  mptiffisupp  32702  mthmval  35580  ec0  38370  0he  43795  limsup0  45709  0cnf  45892