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Theorem fnimaeq0 5333
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 4986 . 2 ((𝐹𝐵) = ∅ ↔ (dom 𝐹𝐵) = ∅)
2 incom 3327 . . . 4 (dom 𝐹𝐵) = (𝐵 ∩ dom 𝐹)
3 fndm 5311 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43sseq2d 3185 . . . . . 6 (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹𝐵𝐴))
54biimpar 297 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ⊆ dom 𝐹)
6 df-ss 3142 . . . . 5 (𝐵 ⊆ dom 𝐹 ↔ (𝐵 ∩ dom 𝐹) = 𝐵)
75, 6sylib 122 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐵 ∩ dom 𝐹) = 𝐵)
82, 7eqtrid 2222 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (dom 𝐹𝐵) = 𝐵)
98eqeq1d 2186 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((dom 𝐹𝐵) = ∅ ↔ 𝐵 = ∅))
101, 9bitrid 192 1 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = ∅ ↔ 𝐵 = ∅))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  cin 3128  wss 3129  c0 3422  dom cdm 4623  cima 4626   Fn wfn 5207
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4629  df-cnv 4631  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-fn 5215
This theorem is referenced by: (None)
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