Theorem 0ima 5640

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 5635	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5532	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3778	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4115	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3760	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1632  ∅c0 4058  ran crn 5267   “ cima 5269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279

This theorem is referenced by:  csbrn  5754  nghmfval  22727  isnghm  22728  mthmval  31779  ec0  34454  0he  38578  limsup0  40429  0cnf  40593  mbf0  40676