Theorem ima0 5144

Description: Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)

Assertion

Ref	Expression
ima0	⊢ (𝐴 “ ∅) = ∅

Proof of Theorem ima0

Step	Hyp	Ref	Expression
1		df-ima 4785	. 2 ⊢ (𝐴 “ ∅) = ran (𝐴 ↾ ∅)
2		res0 5065	. . 3 ⊢ (𝐴 ↾ ∅) = ∅
3	2	rneqi 5008	. 2 ⊢ ran (𝐴 ↾ ∅) = ran ∅
4		rn0 5036	. 2 ⊢ ran ∅ = ∅
5	1, 3, 4	3eqtri 2263	1 ⊢ (𝐴 “ ∅) = ∅

Syntax hints:   = wceq 1402  ∅c0 3520  ran crn 4773   ↾ cres 4774   “ cima 4775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-xp 4778  df-cnv 4780  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785

This theorem is referenced by:  supp0cosupp0fn  6501  fiintim  7232  fidcenumlemrk  7265  fidcenumlemr  7266  ennnfonelem1  13281  ennnfonelemhf1o  13287  eupth2lembfi  16701