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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 5886 . . . . . . . . . 10 (𝐹 = ∅ → 𝐹 = ∅)
6 cnv0 6162 . . . . . . . . . 10 ∅ = ∅
75, 6eqtrdi 2790 . . . . . . . . 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 5933 . . . . 5 dom ∅ = ∅
1311, 12eqtr3di 2789 . . . 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 2734 . . . . . 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 1536  c0 4338  ccnv 5687  dom cdm 5688   Fn wfn 6557  1-1-ontowf1o 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-mo 2537  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569
This theorem is referenced by:  fo00  6884  f1o0  6885  en0  9056  en0ALT  9057  en0r  9058  infn0  9337  symgbas0  19420  derang0  35153  poimirlem28  37634
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