Theorem f0dom0 6726

Description: A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.)

Assertion

Ref	Expression
f0dom0	⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅))

Proof of Theorem f0dom0

Step	Hyp	Ref	Expression
1		feq2 6649	. . . 4 ⊢ (𝑋 = ∅ → (𝐹:𝑋⟶𝑌 ↔ 𝐹:∅⟶𝑌))
2		f0bi 6725	. . . . 5 ⊢ (𝐹:∅⟶𝑌 ↔ 𝐹 = ∅)
3	2	biimpi 216	. . . 4 ⊢ (𝐹:∅⟶𝑌 → 𝐹 = ∅)
4	1, 3	biimtrdi 253	. . 3 ⊢ (𝑋 = ∅ → (𝐹:𝑋⟶𝑌 → 𝐹 = ∅))
5	4	com12 32	. 2 ⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ → 𝐹 = ∅))
6		feq1 6648	. . . 4 ⊢ (𝐹 = ∅ → (𝐹:𝑋⟶𝑌 ↔ ∅:𝑋⟶𝑌))
7		fdm 6679	. . . . 5 ⊢ (∅:𝑋⟶𝑌 → dom ∅ = 𝑋)
8		dm0 5877	. . . . 5 ⊢ dom ∅ = ∅
9	7, 8	eqtr3di 2787	. . . 4 ⊢ (∅:𝑋⟶𝑌 → 𝑋 = ∅)
10	6, 9	biimtrdi 253	. . 3 ⊢ (𝐹 = ∅ → (𝐹:𝑋⟶𝑌 → 𝑋 = ∅))
11	10	com12 32	. 2 ⊢ (𝐹:𝑋⟶𝑌 → (𝐹 = ∅ → 𝑋 = ∅))
12	5, 11	impbid 212	1 ⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542  ∅c0 4287  dom cdm 5632  ⟶wf 6496

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-fun 6502  df-fn 6503  df-f 6504

This theorem is referenced by:  pfxn0  14622  elfrlmbasn0  21730  mavmulsolcl  22507  wrdpmtrlast  33186  fdomne0  49206