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Theorem f1o00 5104
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 5077 . 2 (𝐹:∅–1-1-ontoA ↔ (𝐹 Fn ∅ 𝐹 Fn A))
2 fn0 4961 . . . . . 6 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
32biimpi 113 . . . . 5 (𝐹 Fn ∅ → 𝐹 = ∅)
43adantr 261 . . . 4 ((𝐹 Fn ∅ 𝐹 Fn A) → 𝐹 = ∅)
5 dm0 4492 . . . . 5 dom ∅ = ∅
6 cnveq 4452 . . . . . . . . . 10 (𝐹 = ∅ → 𝐹 = ∅)
7 cnv0 4670 . . . . . . . . . 10 ∅ = ∅
86, 7syl6eq 2085 . . . . . . . . 9 (𝐹 = ∅ → 𝐹 = ∅)
92, 8sylbi 114 . . . . . . . 8 (𝐹 Fn ∅ → 𝐹 = ∅)
109fneq1d 4932 . . . . . . 7 (𝐹 Fn ∅ → (𝐹 Fn A ↔ ∅ Fn A))
1110biimpa 280 . . . . . 6 ((𝐹 Fn ∅ 𝐹 Fn A) → ∅ Fn A)
12 fndm 4941 . . . . . 6 (∅ Fn A → dom ∅ = A)
1311, 12syl 14 . . . . 5 ((𝐹 Fn ∅ 𝐹 Fn A) → dom ∅ = A)
145, 13syl5reqr 2084 . . . 4 ((𝐹 Fn ∅ 𝐹 Fn A) → A = ∅)
154, 14jca 290 . . 3 ((𝐹 Fn ∅ 𝐹 Fn A) → (𝐹 = ∅ A = ∅))
162biimpri 124 . . . . 5 (𝐹 = ∅ → 𝐹 Fn ∅)
1716adantr 261 . . . 4 ((𝐹 = ∅ A = ∅) → 𝐹 Fn ∅)
18 eqid 2037 . . . . . 6 ∅ = ∅
19 fn0 4961 . . . . . 6 (∅ Fn ∅ ↔ ∅ = ∅)
2018, 19mpbir 134 . . . . 5 ∅ Fn ∅
218fneq1d 4932 . . . . . 6 (𝐹 = ∅ → (𝐹 Fn A ↔ ∅ Fn A))
22 fneq2 4931 . . . . . 6 (A = ∅ → (∅ Fn A ↔ ∅ Fn ∅))
2321, 22sylan9bb 435 . . . . 5 ((𝐹 = ∅ A = ∅) → (𝐹 Fn A ↔ ∅ Fn ∅))
2420, 23mpbiri 157 . . . 4 ((𝐹 = ∅ A = ∅) → 𝐹 Fn A)
2517, 24jca 290 . . 3 ((𝐹 = ∅ A = ∅) → (𝐹 Fn ∅ 𝐹 Fn A))
2615, 25impbii 117 . 2 ((𝐹 Fn ∅ 𝐹 Fn A) ↔ (𝐹 = ∅ A = ∅))
271, 26bitri 173 1 (𝐹:∅–1-1-ontoA ↔ (𝐹 = ∅ A = ∅))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  c0 3218  ccnv 4287  dom cdm 4288   Fn wfn 4840  1-1-ontowf1o 4844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852
This theorem is referenced by:  fo00  5105  f1o0  5106  en0  6211
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