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Theorem f1o00 6353
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 6327 . 2 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴))
2 fn0 6188 . . . . . 6 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
32biimpi 207 . . . . 5 (𝐹 Fn ∅ → 𝐹 = ∅)
43adantr 472 . . . 4 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → 𝐹 = ∅)
5 dm0 5506 . . . . 5 dom ∅ = ∅
6 cnveq 5463 . . . . . . . . . 10 (𝐹 = ∅ → 𝐹 = ∅)
7 cnv0 5717 . . . . . . . . . 10 ∅ = ∅
86, 7syl6eq 2814 . . . . . . . . 9 (𝐹 = ∅ → 𝐹 = ∅)
92, 8sylbi 208 . . . . . . . 8 (𝐹 Fn ∅ → 𝐹 = ∅)
109fneq1d 6158 . . . . . . 7 (𝐹 Fn ∅ → (𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
1110biimpa 468 . . . . . 6 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → ∅ Fn 𝐴)
12 fndm 6167 . . . . . 6 (∅ Fn 𝐴 → dom ∅ = 𝐴)
1311, 12syl 17 . . . . 5 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → dom ∅ = 𝐴)
145, 13syl5reqr 2813 . . . 4 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → 𝐴 = ∅)
154, 14jca 507 . . 3 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → (𝐹 = ∅ ∧ 𝐴 = ∅))
162biimpri 219 . . . . 5 (𝐹 = ∅ → 𝐹 Fn ∅)
1716adantr 472 . . . 4 ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn ∅)
18 eqid 2764 . . . . . 6 ∅ = ∅
19 fn0 6188 . . . . . 6 (∅ Fn ∅ ↔ ∅ = ∅)
2018, 19mpbir 222 . . . . 5 ∅ Fn ∅
218fneq1d 6158 . . . . . 6 (𝐹 = ∅ → (𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
22 fneq2 6157 . . . . . 6 (𝐴 = ∅ → (∅ Fn 𝐴 ↔ ∅ Fn ∅))
2321, 22sylan9bb 505 . . . . 5 ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn 𝐴 ↔ ∅ Fn ∅))
2420, 23mpbiri 249 . . . 4 ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn 𝐴)
2517, 24jca 507 . . 3 ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴))
2615, 25impbii 200 . 2 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
271, 26bitri 266 1 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  c0 4078  ccnv 5275  dom cdm 5276   Fn wfn 6062  1-1-ontowf1o 6066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pr 5061
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-br 4809  df-opab 4871  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074
This theorem is referenced by:  fo00  6354  f1o0  6355  en0  8222  symgbas0  18078  derang0  31530  poimirlem28  33793
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