Theorem 0ima 5788

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 5783	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5677	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3895	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4237	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3876	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1507  ∅c0 4180  ran crn 5409   “ cima 5411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pr 5187

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-br 4931  df-opab 4993  df-xp 5414  df-cnv 5416  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421

This theorem is referenced by:  csbrn  5901  nghmfval  23037  isnghm  23038  mthmval  32342  ec0  35066  0he  39491  limsup0  41407  0cnf  41591