Theorem 0ima 6066

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 6060	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5917	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 4014	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4392	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3994	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1541  ∅c0 4318  ran crn 5670   “ cima 5672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682

This theorem is referenced by:  csbrn  6191  nghmfval  24168  isnghm  24169  mptiffisupp  31786  mthmval  34397  ec0  37043  0he  42304  limsup0  44183  0cnf  44366