Theorem 0ima 6049

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 6042	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5889	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3995	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4363	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3963	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1540  ∅c0 4296  ran crn 5639   “ cima 5641

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 5251  ax-nul 5261  ax-pr 5387

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 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651

This theorem is referenced by:  csbrn  6176  nghmfval  24610  isnghm  24611  mptiffisupp  32616  mthmval  35562  ec0  38351  0he  43771  limsup0  45692  0cnf  45875