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Theorem f0dom0 6127
Description: A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.)
Assertion
Ref Expression
f0dom0 (𝐹:𝑋𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅))

Proof of Theorem f0dom0
StepHypRef Expression
1 feq2 6065 . . . 4 (𝑋 = ∅ → (𝐹:𝑋𝑌𝐹:∅⟶𝑌))
2 f0bi 6126 . . . . 5 (𝐹:∅⟶𝑌𝐹 = ∅)
32biimpi 206 . . . 4 (𝐹:∅⟶𝑌𝐹 = ∅)
41, 3syl6bi 243 . . 3 (𝑋 = ∅ → (𝐹:𝑋𝑌𝐹 = ∅))
54com12 32 . 2 (𝐹:𝑋𝑌 → (𝑋 = ∅ → 𝐹 = ∅))
6 feq1 6064 . . . 4 (𝐹 = ∅ → (𝐹:𝑋𝑌 ↔ ∅:𝑋𝑌))
7 dm0 5371 . . . . 5 dom ∅ = ∅
8 fdm 6089 . . . . 5 (∅:𝑋𝑌 → dom ∅ = 𝑋)
97, 8syl5reqr 2700 . . . 4 (∅:𝑋𝑌𝑋 = ∅)
106, 9syl6bi 243 . . 3 (𝐹 = ∅ → (𝐹:𝑋𝑌𝑋 = ∅))
1110com12 32 . 2 (𝐹:𝑋𝑌 → (𝐹 = ∅ → 𝑋 = ∅))
125, 11impbid 202 1 (𝐹:𝑋𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  c0 3948  dom cdm 5143  wf 5922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-fun 5928  df-fn 5929  df-f 5930
This theorem is referenced by:  swrdn0  13476  elfrlmbasn0  20154  mavmulsolcl  20405
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