Theorem 0ima 6075

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 6068	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5922	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 4014	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4392	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3994	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1534  ∅c0 4318  ran crn 5673   “ cima 5675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685

This theorem is referenced by:  csbrn  6201  nghmfval  24626  isnghm  24627  mptiffisupp  32457  mthmval  35121  ec0  37777  0he  43135  limsup0  45005  0cnf  45188