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Theorem fo00 6860
Description: Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
Assertion
Ref Expression
fo00 (𝐹:∅–onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))

Proof of Theorem fo00
StepHypRef Expression
1 fofn 6798 . . . . . 6 (𝐹:∅–onto𝐴𝐹 Fn ∅)
2 fn0 6672 . . . . . . 7 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
3 f10 6857 . . . . . . . 8 ∅:∅–1-1𝐴
4 f1eq1 6773 . . . . . . . 8 (𝐹 = ∅ → (𝐹:∅–1-1𝐴 ↔ ∅:∅–1-1𝐴))
53, 4mpbiri 258 . . . . . . 7 (𝐹 = ∅ → 𝐹:∅–1-1𝐴)
62, 5sylbi 216 . . . . . 6 (𝐹 Fn ∅ → 𝐹:∅–1-1𝐴)
71, 6syl 17 . . . . 5 (𝐹:∅–onto𝐴𝐹:∅–1-1𝐴)
87ancri 549 . . . 4 (𝐹:∅–onto𝐴 → (𝐹:∅–1-1𝐴𝐹:∅–onto𝐴))
9 df-f1o 6541 . . . 4 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹:∅–1-1𝐴𝐹:∅–onto𝐴))
108, 9sylibr 233 . . 3 (𝐹:∅–onto𝐴𝐹:∅–1-1-onto𝐴)
11 f1ofo 6831 . . 3 (𝐹:∅–1-1-onto𝐴𝐹:∅–onto𝐴)
1210, 11impbii 208 . 2 (𝐹:∅–onto𝐴𝐹:∅–1-1-onto𝐴)
13 f1o00 6859 . 2 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
1412, 13bitri 275 1 (𝐹:∅–onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  c0 4315   Fn wfn 6529  1-1wf1 6531  ontowfo 6532  1-1-ontowf1o 6533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541
This theorem is referenced by:  fsumf1o  15667  fprodf1o  15888  0ramcl  16957  fullthinc  47878
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