Theorem 0ima 6085

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 6078	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5921	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3988	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4360	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3956	1 ⊢ (∅ “ 𝐴) = ∅

This proof depends on syntax axioms:   = wceq 1570  ∅c0 4289  ran crn 5667   “ cima 5669

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 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

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 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679

This theorem is used by:  csbrn  6209  nghmfval  24916  isnghm  24917  mptiffisupp  33075  vieta  34001  mthmval  36088  ec0  39067  0he  44549  limsup0  46449  0cnf  46632