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Theorem fnimaeq0 4963
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 A BA) → ((𝐹B) = ∅ ↔ B = ∅))

Proof of Theorem fnimaeq0
StepHypRef Expression
1 imadisj 4630 . 2 ((𝐹B) = ∅ ↔ (dom 𝐹B) = ∅)
2 incom 3123 . . . 4 (dom 𝐹B) = (B ∩ dom 𝐹)
3 fndm 4941 . . . . . . 7 (𝐹 Fn A → dom 𝐹 = A)
43sseq2d 2967 . . . . . 6 (𝐹 Fn A → (B ⊆ dom 𝐹BA))
54biimpar 281 . . . . 5 ((𝐹 Fn A BA) → B ⊆ dom 𝐹)
6 df-ss 2925 . . . . 5 (B ⊆ dom 𝐹 ↔ (B ∩ dom 𝐹) = B)
75, 6sylib 127 . . . 4 ((𝐹 Fn A BA) → (B ∩ dom 𝐹) = B)
82, 7syl5eq 2081 . . 3 ((𝐹 Fn A BA) → (dom 𝐹B) = B)
98eqeq1d 2045 . 2 ((𝐹 Fn A BA) → ((dom 𝐹B) = ∅ ↔ B = ∅))
101, 9syl5bb 181 1 ((𝐹 Fn A BA) → ((𝐹B) = ∅ ↔ B = ∅))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  cin 2910  wss 2911  c0 3218  dom cdm 4288  cima 4291   Fn wfn 4840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fn 4848
This theorem is referenced by: (None)
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