Theorem 0ima 6071

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 6064	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5907	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3987	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4357	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3955	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1563  ∅c0 4288  ran crn 5653   “ cima 5655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-cnv 5660  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665

This theorem is referenced by:  csbrn  6194  nghmfval  24840  isnghm  24841  mptiffisupp  32950  vieta  33887  mthmval  35938  ec0  38888  0he  44370  limsup0  46266  0cnf  46449