Theorem 0ima 6200

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 6197	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5907	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3956	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4349	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3924	1 ⊢ (∅ “ 𝐴) = ∅

This proof depends on syntax axioms:   = wceq 1569  ∅c0 4278  ran crn 5651   “ cima 5653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663

This theorem is used by:  csbrn  6204  nghmfval  25046  isnghm  25047  mptiffisupp  33333  vieta  34258  mthmval  36426  ec0  39402  0he  44875  limsup0  46797  0cnf  46980