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Theorem f1o00 6624
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 6598 . 2 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴))
2 fn0 6451 . . . . . 6 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
32biimpi 219 . . . . 5 (𝐹 Fn ∅ → 𝐹 = ∅)
43adantr 484 . . . 4 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → 𝐹 = ∅)
5 dm0 5754 . . . . 5 dom ∅ = ∅
6 cnveq 5708 . . . . . . . . . 10 (𝐹 = ∅ → 𝐹 = ∅)
7 cnv0 5966 . . . . . . . . . 10 ∅ = ∅
86, 7eqtrdi 2849 . . . . . . . . 9 (𝐹 = ∅ → 𝐹 = ∅)
92, 8sylbi 220 . . . . . . . 8 (𝐹 Fn ∅ → 𝐹 = ∅)
109fneq1d 6416 . . . . . . 7 (𝐹 Fn ∅ → (𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
1110biimpa 480 . . . . . 6 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → ∅ Fn 𝐴)
1211fndmd 6427 . . . . 5 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → dom ∅ = 𝐴)
135, 12syl5reqr 2848 . . . 4 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → 𝐴 = ∅)
144, 13jca 515 . . 3 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → (𝐹 = ∅ ∧ 𝐴 = ∅))
152biimpri 231 . . . . 5 (𝐹 = ∅ → 𝐹 Fn ∅)
1615adantr 484 . . . 4 ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn ∅)
17 eqid 2798 . . . . . 6 ∅ = ∅
18 fn0 6451 . . . . . 6 (∅ Fn ∅ ↔ ∅ = ∅)
1917, 18mpbir 234 . . . . 5 ∅ Fn ∅
208fneq1d 6416 . . . . . 6 (𝐹 = ∅ → (𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
21 fneq2 6415 . . . . . 6 (𝐴 = ∅ → (∅ Fn 𝐴 ↔ ∅ Fn ∅))
2220, 21sylan9bb 513 . . . . 5 ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn 𝐴 ↔ ∅ Fn ∅))
2319, 22mpbiri 261 . . . 4 ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn 𝐴)
2416, 23jca 515 . . 3 ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴))
2514, 24impbii 212 . 2 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
261, 25bitri 278 1 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  c0 4243  ccnv 5518  dom cdm 5519   Fn wfn 6319  1-1-ontowf1o 6323
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331
This theorem is referenced by:  fo00  6625  f1o0  6626  en0  8555  symgbas0  18509  derang0  32526  poimirlem28  35082
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