ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ima0 GIF version

Theorem ima0 4712
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
StepHypRef Expression
1 df-ima 4386 . 2 (𝐴 “ ∅) = ran (𝐴 ↾ ∅)
2 res0 4644 . . 3 (𝐴 ↾ ∅) = ∅
32rneqi 4590 . 2 ran (𝐴 ↾ ∅) = ran ∅
4 rn0 4616 . 2 ran ∅ = ∅
51, 3, 43eqtri 2080 1 (𝐴 “ ∅) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1259  c0 3252  ran crn 4374  cres 4375  cima 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator