Theorem 0ima 6033

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 6026	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5872	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3986	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4353	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3954	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1540  ∅c0 4286  ran crn 5624   “ cima 5626

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636

This theorem is referenced by:  csbrn  6156  nghmfval  24626  isnghm  24627  mptiffisupp  32649  mthmval  35547  ec0  38336  0he  43755  limsup0  45676  0cnf  45859