Theorem 0ima 6074

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 6067	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5910	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3979	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4350	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3947	1 ⊢ (∅ “ 𝐴) = ∅

This proof depends on syntax axioms:   = wceq 1570  ∅c0 4279  ran crn 5656   “ cima 5658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668

This theorem is used by:  csbrn  6199  nghmfval  24951  isnghm  24952  mptiffisupp  33168  vieta  34093  mthmval  36157  ec0  39128  0he  44625  limsup0  46525  0cnf  46708