Theorem 0ima 6064

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 6057	. . 3 ⊢ (∅ “ 𝐴) ⊆ ran ∅
2		rn0 5900	. . 3 ⊢ ran ∅ = ∅
3	1, 2	sseqtri 3984	. 2 ⊢ (∅ “ 𝐴) ⊆ ∅
4		0ss 4353	. 2 ⊢ ∅ ⊆ (∅ “ 𝐴)
5	3, 4	eqssi 3952	1 ⊢ (∅ “ 𝐴) = ∅

Syntax hints:   = wceq 1559  ∅c0 4285  ran crn 5646   “ cima 5648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-cnv 5653  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658

This theorem is referenced by:  csbrn  6186  nghmfval  24762  isnghm  24763  mptiffisupp  32845  vieta  33838  mthmval  35889  ec0  38840  0he  44322  limsup0  46232  0cnf  46415