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Theorem fnimaeq0 6683
Description: Images under a function never map nonempty sets to empty sets. EDITORIAL: usable in fnwe2lem2 41783. (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 6079 . 2 ((𝐹𝐵) = ∅ ↔ (dom 𝐹𝐵) = ∅)
2 incom 4201 . . . 4 (dom 𝐹𝐵) = (𝐵 ∩ dom 𝐹)
3 fndm 6652 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43sseq2d 4014 . . . . . 6 (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹𝐵𝐴))
54biimpar 478 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ⊆ dom 𝐹)
6 df-ss 3965 . . . . 5 (𝐵 ⊆ dom 𝐹 ↔ (𝐵 ∩ dom 𝐹) = 𝐵)
75, 6sylib 217 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐵 ∩ dom 𝐹) = 𝐵)
82, 7eqtrid 2784 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (dom 𝐹𝐵) = 𝐵)
98eqeq1d 2734 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((dom 𝐹𝐵) = ∅ ↔ 𝐵 = ∅))
101, 9bitrid 282 1 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = ∅ ↔ 𝐵 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  cin 3947  wss 3948  c0 4322  dom cdm 5676  cima 5679   Fn wfn 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-fn 6546
This theorem is referenced by:  ipodrsima  18493  mdegldg  25583  ig1peu  25688  ig1pdvds  25693  dimval  32681  dimvalfi  32682  nummin  34089  kelac1  41795
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