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Theorem f00 5024
Description: A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.)
Assertion
Ref Expression
f00 (𝐹:A⟶∅ ↔ (𝐹 = ∅ A = ∅))

Proof of Theorem f00
StepHypRef Expression
1 ffun 4991 . . . . 5 (𝐹:A⟶∅ → Fun 𝐹)
2 frn 4995 . . . . . . 7 (𝐹:A⟶∅ → ran 𝐹 ⊆ ∅)
3 ss0 3251 . . . . . . 7 (ran 𝐹 ⊆ ∅ → ran 𝐹 = ∅)
42, 3syl 14 . . . . . 6 (𝐹:A⟶∅ → ran 𝐹 = ∅)
5 dm0rn0 4495 . . . . . 6 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
64, 5sylibr 137 . . . . 5 (𝐹:A⟶∅ → dom 𝐹 = ∅)
7 df-fn 4848 . . . . 5 (𝐹 Fn ∅ ↔ (Fun 𝐹 dom 𝐹 = ∅))
81, 6, 7sylanbrc 394 . . . 4 (𝐹:A⟶∅ → 𝐹 Fn ∅)
9 fn0 4961 . . . 4 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
108, 9sylib 127 . . 3 (𝐹:A⟶∅ → 𝐹 = ∅)
11 fdm 4993 . . . 4 (𝐹:A⟶∅ → dom 𝐹 = A)
1211, 6eqtr3d 2071 . . 3 (𝐹:A⟶∅ → A = ∅)
1310, 12jca 290 . 2 (𝐹:A⟶∅ → (𝐹 = ∅ A = ∅))
14 f0 5023 . . 3 ∅:∅⟶∅
15 feq1 4973 . . . 4 (𝐹 = ∅ → (𝐹:A⟶∅ ↔ ∅:A⟶∅))
16 feq2 4974 . . . 4 (A = ∅ → (∅:A⟶∅ ↔ ∅:∅⟶∅))
1715, 16sylan9bb 435 . . 3 ((𝐹 = ∅ A = ∅) → (𝐹:A⟶∅ ↔ ∅:∅⟶∅))
1814, 17mpbiri 157 . 2 ((𝐹 = ∅ A = ∅) → 𝐹:A⟶∅)
1913, 18impbii 117 1 (𝐹:A⟶∅ ↔ (𝐹 = ∅ A = ∅))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  wss 2911  c0 3218  dom cdm 4288  ran crn 4289  Fun wfun 4839   Fn wfn 4840  wf 4841
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-nul 3874  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-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849
This theorem is referenced by: (None)
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