Theorem 0ima 6082

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 6075	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5918	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3986	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4358	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3954	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1570  ∅c0 4287  ran crn 5664   “ cima 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is referenced by:  csbrn  6206  nghmfval  24860  isnghm  24861  mptiffisupp  33016  vieta  33948  mthmval  36045  ec0  39004  0he  44488  limsup0  46388  0cnf  46571