MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1o00 Structured version   Visualization version   GIF version

Theorem f1o00 6883
Description: One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
Assertion
Ref Expression
f1o00 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))

Proof of Theorem f1o00
StepHypRef Expression
1 dff1o4 6856 . 2 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴))
2 fn0 6699 . . . . . 6 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
32biimpi 216 . . . . 5 (𝐹 Fn ∅ → 𝐹 = ∅)
43adantr 480 . . . 4 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → 𝐹 = ∅)
5 cnveq 5884 . . . . . . . . . 10 (𝐹 = ∅ → 𝐹 = ∅)
6 cnv0 6160 . . . . . . . . . 10 ∅ = ∅
75, 6eqtrdi 2793 . . . . . . . . 9 (𝐹 = ∅ → 𝐹 = ∅)
82, 7sylbi 217 . . . . . . . 8 (𝐹 Fn ∅ → 𝐹 = ∅)
98fneq1d 6661 . . . . . . 7 (𝐹 Fn ∅ → (𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
109biimpa 476 . . . . . 6 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → ∅ Fn 𝐴)
1110fndmd 6673 . . . . 5 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → dom ∅ = 𝐴)
12 dm0 5931 . . . . 5 dom ∅ = ∅
1311, 12eqtr3di 2792 . . . 4 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → 𝐴 = ∅)
144, 13jca 511 . . 3 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → (𝐹 = ∅ ∧ 𝐴 = ∅))
152biimpri 228 . . . . 5 (𝐹 = ∅ → 𝐹 Fn ∅)
1615adantr 480 . . . 4 ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn ∅)
17 eqid 2737 . . . . . 6 ∅ = ∅
18 fn0 6699 . . . . . 6 (∅ Fn ∅ ↔ ∅ = ∅)
1917, 18mpbir 231 . . . . 5 ∅ Fn ∅
207fneq1d 6661 . . . . . 6 (𝐹 = ∅ → (𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
21 fneq2 6660 . . . . . 6 (𝐴 = ∅ → (∅ Fn 𝐴 ↔ ∅ Fn ∅))
2220, 21sylan9bb 509 . . . . 5 ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn 𝐴 ↔ ∅ Fn ∅))
2319, 22mpbiri 258 . . . 4 ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn 𝐴)
2416, 23jca 511 . . 3 ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴))
2514, 24impbii 209 . 2 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
261, 25bitri 275 1 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  c0 4333  ccnv 5684  dom cdm 5685   Fn wfn 6556  1-1-ontowf1o 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568
This theorem is referenced by:  fo00  6884  f1o0  6885  en0  9058  en0ALT  9059  en0r  9060  infn0  9340  symgbas0  19406  derang0  35174  poimirlem28  37655
  Copyright terms: Public domain W3C validator