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Theorem f00 6547
 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 6502 . . . . 5 (𝐹:𝐴⟶∅ → Fun 𝐹)
2 frn 6505 . . . . . . 7 (𝐹:𝐴⟶∅ → ran 𝐹 ⊆ ∅)
3 ss0 4295 . . . . . . 7 (ran 𝐹 ⊆ ∅ → ran 𝐹 = ∅)
42, 3syl 17 . . . . . 6 (𝐹:𝐴⟶∅ → ran 𝐹 = ∅)
5 dm0rn0 5767 . . . . . 6 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
64, 5sylibr 237 . . . . 5 (𝐹:𝐴⟶∅ → dom 𝐹 = ∅)
7 df-fn 6339 . . . . 5 (𝐹 Fn ∅ ↔ (Fun 𝐹 ∧ dom 𝐹 = ∅))
81, 6, 7sylanbrc 587 . . . 4 (𝐹:𝐴⟶∅ → 𝐹 Fn ∅)
9 fn0 6463 . . . 4 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
108, 9sylib 221 . . 3 (𝐹:𝐴⟶∅ → 𝐹 = ∅)
11 fdm 6507 . . . 4 (𝐹:𝐴⟶∅ → dom 𝐹 = 𝐴)
1211, 6eqtr3d 2796 . . 3 (𝐹:𝐴⟶∅ → 𝐴 = ∅)
1310, 12jca 516 . 2 (𝐹:𝐴⟶∅ → (𝐹 = ∅ ∧ 𝐴 = ∅))
14 f0 6546 . . 3 ∅:∅⟶∅
15 feq1 6480 . . . 4 (𝐹 = ∅ → (𝐹:𝐴⟶∅ ↔ ∅:𝐴⟶∅))
16 feq2 6481 . . . 4 (𝐴 = ∅ → (∅:𝐴⟶∅ ↔ ∅:∅⟶∅))
1715, 16sylan9bb 514 . . 3 ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹:𝐴⟶∅ ↔ ∅:∅⟶∅))
1814, 17mpbiri 261 . 2 ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹:𝐴⟶∅)
1913, 18impbii 212 1 (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1539   ⊆ wss 3859  ∅c0 4226  dom cdm 5525  ran crn 5526  Fun wfun 6330   Fn wfn 6331  ⟶wf 6332 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-opab 5096  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-fun 6338  df-fn 6339  df-f 6340 This theorem is referenced by:  cantnff  9163  0wrd0  13932  supcvg  15252  ram0  16406  itgsubstlem  24740  uhgr0vb  26957  lfuhgr1v0e  27136  wlkv0  27532  sate0fv0  32888  prv0  32901  ismgmOLD  35561
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