Theorem 0ima 6037

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 6030	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5875	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3970	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4335	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3938	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1547  ∅c0 4268  ran crn 5626   “ cima 5628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638

This theorem is referenced by:  csbrn  6161  nghmfval  24712  isnghm  24713  mptiffisupp  32792  vieta  33771  mthmval  35810  ec0  38751  0he  44233  limsup0  46144  0cnf  46327