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Theorem f1o00 6865
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 6838 . 2 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴))
2 fn0 6678 . . . . . 6 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
32biimpi 215 . . . . 5 (𝐹 Fn ∅ → 𝐹 = ∅)
43adantr 482 . . . 4 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → 𝐹 = ∅)
5 cnveq 5871 . . . . . . . . . 10 (𝐹 = ∅ → 𝐹 = ∅)
6 cnv0 6137 . . . . . . . . . 10 ∅ = ∅
75, 6eqtrdi 2789 . . . . . . . . 9 (𝐹 = ∅ → 𝐹 = ∅)
82, 7sylbi 216 . . . . . . . 8 (𝐹 Fn ∅ → 𝐹 = ∅)
98fneq1d 6639 . . . . . . 7 (𝐹 Fn ∅ → (𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
109biimpa 478 . . . . . 6 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → ∅ Fn 𝐴)
1110fndmd 6651 . . . . 5 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → dom ∅ = 𝐴)
12 dm0 5918 . . . . 5 dom ∅ = ∅
1311, 12eqtr3di 2788 . . . 4 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → 𝐴 = ∅)
144, 13jca 513 . . 3 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → (𝐹 = ∅ ∧ 𝐴 = ∅))
152biimpri 227 . . . . 5 (𝐹 = ∅ → 𝐹 Fn ∅)
1615adantr 482 . . . 4 ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn ∅)
17 eqid 2733 . . . . . 6 ∅ = ∅
18 fn0 6678 . . . . . 6 (∅ Fn ∅ ↔ ∅ = ∅)
1917, 18mpbir 230 . . . . 5 ∅ Fn ∅
207fneq1d 6639 . . . . . 6 (𝐹 = ∅ → (𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
21 fneq2 6638 . . . . . 6 (𝐴 = ∅ → (∅ Fn 𝐴 ↔ ∅ Fn ∅))
2220, 21sylan9bb 511 . . . . 5 ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn 𝐴 ↔ ∅ Fn ∅))
2319, 22mpbiri 258 . . . 4 ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn 𝐴)
2416, 23jca 513 . . 3 ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴))
2514, 24impbii 208 . 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 205  wa 397   = wceq 1542  c0 4321  ccnv 5674  dom cdm 5675   Fn wfn 6535  1-1-ontowf1o 6539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547
This theorem is referenced by:  fo00  6866  f1o0  6867  en0  9009  en0OLD  9010  en0ALT  9011  en0r  9012  infn0  9303  symgbas0  19249  derang0  34098  poimirlem28  36454
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