Theorem 0ima 6026

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 6019	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5865	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3978	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4347	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3946	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1541  ∅c0 4280  ran crn 5615   “ cima 5617

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

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 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627

This theorem is referenced by:  csbrn  6150  nghmfval  24637  isnghm  24638  mptiffisupp  32674  mthmval  35619  ec0  38400  0he  43874  limsup0  45791  0cnf  45974