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Theorem fnimaeq0 6701
Description: Images under a function never map nonempty sets to empty sets. EDITORIAL: usable in fnwe2lem2 43063. (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 6098 . 2 ((𝐹𝐵) = ∅ ↔ (dom 𝐹𝐵) = ∅)
2 incom 4209 . . . 4 (dom 𝐹𝐵) = (𝐵 ∩ dom 𝐹)
3 fndm 6671 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43sseq2d 4016 . . . . . 6 (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹𝐵𝐴))
54biimpar 477 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ⊆ dom 𝐹)
6 dfss2 3969 . . . . 5 (𝐵 ⊆ dom 𝐹 ↔ (𝐵 ∩ dom 𝐹) = 𝐵)
75, 6sylib 218 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐵 ∩ dom 𝐹) = 𝐵)
82, 7eqtrid 2789 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (dom 𝐹𝐵) = 𝐵)
98eqeq1d 2739 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((dom 𝐹𝐵) = ∅ ↔ 𝐵 = ∅))
101, 9bitrid 283 1 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = ∅ ↔ 𝐵 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  cin 3950  wss 3951  c0 4333  dom cdm 5685  cima 5688   Fn wfn 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fn 6564
This theorem is referenced by:  ipodrsima  18586  mdegldg  26105  ig1peu  26214  ig1pdvds  26219  dimval  33651  dimvalfi  33652  nummin  35105  aks6d1c6lem3  42173  kelac1  43075
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