Theorem 0ima 6065

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 6058	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5905	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 4007	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4375	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3975	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1540  ∅c0 4308  ran crn 5655   “ cima 5657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667

This theorem is referenced by:  csbrn  6192  nghmfval  24661  isnghm  24662  mptiffisupp  32670  mthmval  35597  ec0  38387  0he  43806  limsup0  45723  0cnf  45906