ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fnimaeq0 GIF version

Theorem fnimaeq0 5200
Description: Images under a function never map nonempty sets to empty sets. (Contributed by Stefan O'Rear, 21-Jan-2015.)
Assertion
Ref Expression
fnimaeq0 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = ∅ ↔ 𝐵 = ∅))

Proof of Theorem fnimaeq0
StepHypRef Expression
1 imadisj 4857 . 2 ((𝐹𝐵) = ∅ ↔ (dom 𝐹𝐵) = ∅)
2 incom 3232 . . . 4 (dom 𝐹𝐵) = (𝐵 ∩ dom 𝐹)
3 fndm 5178 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43sseq2d 3091 . . . . . 6 (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹𝐵𝐴))
54biimpar 293 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ⊆ dom 𝐹)
6 df-ss 3048 . . . . 5 (𝐵 ⊆ dom 𝐹 ↔ (𝐵 ∩ dom 𝐹) = 𝐵)
75, 6sylib 121 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐵 ∩ dom 𝐹) = 𝐵)
82, 7syl5eq 2157 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (dom 𝐹𝐵) = 𝐵)
98eqeq1d 2121 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((dom 𝐹𝐵) = ∅ ↔ 𝐵 = ∅))
101, 9syl5bb 191 1 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = ∅ ↔ 𝐵 = ∅))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1312  cin 3034  wss 3035  c0 3327  dom cdm 4497  cima 4500   Fn wfn 5074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-xp 4503  df-cnv 4505  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-fn 5082
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator