Theorem ima0 5083

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 4729	. 2 ⊢ (𝐴 “ ∅) = ran (𝐴 ↾ ∅)
2		res0 5005	. . 3 ⊢ (𝐴 ↾ ∅) = ∅
3	2	rneqi 4948	. 2 ⊢ ran (𝐴 ↾ ∅) = ran ∅
4		rn0 4976	. 2 ⊢ ran ∅ = ∅
5	1, 3, 4	3eqtri 2254	1 ⊢ (𝐴 “ ∅) = ∅

Syntax hints:   = wceq 1395  ∅c0 3491  ran crn 4717   ↾ cres 4718   “ cima 4719

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4722  df-cnv 4724  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729

This theorem is referenced by:  fiintim  7081  fidcenumlemrk  7109  fidcenumlemr  7110  ennnfonelem1  12964  ennnfonelemhf1o  12970