Theorem 0ima 6031

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 6024	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5870	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3979	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4349	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3947	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1541  ∅c0 4282  ran crn 5620   “ cima 5622

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 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

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 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-xp 5625  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632

This theorem is referenced by:  csbrn  6155  nghmfval  24638  isnghm  24639  mptiffisupp  32678  mthmval  35640  ec0  38422  0he  43900  limsup0  45817  0cnf  46000