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Theorem f1o00 6313
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 6287 . 2 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴))
2 fn0 6150 . . . . . 6 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
32biimpi 206 . . . . 5 (𝐹 Fn ∅ → 𝐹 = ∅)
43adantr 466 . . . 4 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → 𝐹 = ∅)
5 dm0 5476 . . . . 5 dom ∅ = ∅
6 cnveq 5433 . . . . . . . . . 10 (𝐹 = ∅ → 𝐹 = ∅)
7 cnv0 5675 . . . . . . . . . 10 ∅ = ∅
86, 7syl6eq 2821 . . . . . . . . 9 (𝐹 = ∅ → 𝐹 = ∅)
92, 8sylbi 207 . . . . . . . 8 (𝐹 Fn ∅ → 𝐹 = ∅)
109fneq1d 6120 . . . . . . 7 (𝐹 Fn ∅ → (𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
1110biimpa 462 . . . . . 6 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → ∅ Fn 𝐴)
12 fndm 6129 . . . . . 6 (∅ Fn 𝐴 → dom ∅ = 𝐴)
1311, 12syl 17 . . . . 5 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → dom ∅ = 𝐴)
145, 13syl5reqr 2820 . . . 4 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → 𝐴 = ∅)
154, 14jca 501 . . 3 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → (𝐹 = ∅ ∧ 𝐴 = ∅))
162biimpri 218 . . . . 5 (𝐹 = ∅ → 𝐹 Fn ∅)
1716adantr 466 . . . 4 ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn ∅)
18 eqid 2771 . . . . . 6 ∅ = ∅
19 fn0 6150 . . . . . 6 (∅ Fn ∅ ↔ ∅ = ∅)
2018, 19mpbir 221 . . . . 5 ∅ Fn ∅
218fneq1d 6120 . . . . . 6 (𝐹 = ∅ → (𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
22 fneq2 6119 . . . . . 6 (𝐴 = ∅ → (∅ Fn 𝐴 ↔ ∅ Fn ∅))
2321, 22sylan9bb 499 . . . . 5 ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn 𝐴 ↔ ∅ Fn ∅))
2420, 23mpbiri 248 . . . 4 ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn 𝐴)
2517, 24jca 501 . . 3 ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴))
2615, 25impbii 199 . 2 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
271, 26bitri 264 1 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  c0 4063  ccnv 5249  dom cdm 5250   Fn wfn 6025  1-1-ontowf1o 6029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037
This theorem is referenced by:  fo00  6314  f1o0  6315  en0  8176  symgbas0  18021  derang0  31489  poimirlem28  33770
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