Theorem 0ima 6107

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 6100	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5950	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 4045	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4423	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 4025	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1537  ∅c0 4352  ran crn 5701   “ cima 5703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713

This theorem is referenced by:  csbrn  6234  nghmfval  24764  isnghm  24765  mptiffisupp  32705  mthmval  35543  ec0  38325  0he  43744  limsup0  45615  0cnf  45798