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Theorem f1o00 6868
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 6841 . 2 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴))
2 fn0 6681 . . . . . 6 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
32biimpi 215 . . . . 5 (𝐹 Fn ∅ → 𝐹 = ∅)
43adantr 481 . . . 4 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → 𝐹 = ∅)
5 cnveq 5873 . . . . . . . . . 10 (𝐹 = ∅ → 𝐹 = ∅)
6 cnv0 6140 . . . . . . . . . 10 ∅ = ∅
75, 6eqtrdi 2788 . . . . . . . . 9 (𝐹 = ∅ → 𝐹 = ∅)
82, 7sylbi 216 . . . . . . . 8 (𝐹 Fn ∅ → 𝐹 = ∅)
98fneq1d 6642 . . . . . . 7 (𝐹 Fn ∅ → (𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
109biimpa 477 . . . . . 6 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → ∅ Fn 𝐴)
1110fndmd 6654 . . . . 5 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → dom ∅ = 𝐴)
12 dm0 5920 . . . . 5 dom ∅ = ∅
1311, 12eqtr3di 2787 . . . 4 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → 𝐴 = ∅)
144, 13jca 512 . . 3 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → (𝐹 = ∅ ∧ 𝐴 = ∅))
152biimpri 227 . . . . 5 (𝐹 = ∅ → 𝐹 Fn ∅)
1615adantr 481 . . . 4 ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn ∅)
17 eqid 2732 . . . . . 6 ∅ = ∅
18 fn0 6681 . . . . . 6 (∅ Fn ∅ ↔ ∅ = ∅)
1917, 18mpbir 230 . . . . 5 ∅ Fn ∅
207fneq1d 6642 . . . . . 6 (𝐹 = ∅ → (𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
21 fneq2 6641 . . . . . 6 (𝐴 = ∅ → (∅ Fn 𝐴 ↔ ∅ Fn ∅))
2220, 21sylan9bb 510 . . . . 5 ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn 𝐴 ↔ ∅ Fn ∅))
2319, 22mpbiri 257 . . . 4 ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn 𝐴)
2416, 23jca 512 . . 3 ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴))
2514, 24impbii 208 . 2 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
261, 25bitri 274 1 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  c0 4322  ccnv 5675  dom cdm 5676   Fn wfn 6538  1-1-ontowf1o 6542
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-mo 2534  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550
This theorem is referenced by:  fo00  6869  f1o0  6870  en0  9015  en0OLD  9016  en0ALT  9017  en0r  9018  infn0  9309  symgbas0  19297  derang0  34446  poimirlem28  36819
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