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Theorem fo00 6884
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 6822 . . . . . 6 (𝐹:∅–onto𝐴𝐹 Fn ∅)
2 fn0 6699 . . . . . . 7 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
3 f10 6881 . . . . . . . 8 ∅:∅–1-1𝐴
4 f1eq1 6799 . . . . . . . 8 (𝐹 = ∅ → (𝐹:∅–1-1𝐴 ↔ ∅:∅–1-1𝐴))
53, 4mpbiri 258 . . . . . . 7 (𝐹 = ∅ → 𝐹:∅–1-1𝐴)
62, 5sylbi 217 . . . . . 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 6569 . . . 4 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹:∅–1-1𝐴𝐹:∅–onto𝐴))
108, 9sylibr 234 . . 3 (𝐹:∅–onto𝐴𝐹:∅–1-1-onto𝐴)
11 f1ofo 6855 . . 3 (𝐹:∅–1-1-onto𝐴𝐹:∅–onto𝐴)
1210, 11impbii 209 . 2 (𝐹:∅–onto𝐴𝐹:∅–1-1-onto𝐴)
13 f1o00 6883 . 2 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
1412, 13bitri 275 1 (𝐹:∅–onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1536  c0 4338   Fn wfn 6557  1-1wf1 6559  ontowfo 6560  1-1-ontowf1o 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569
This theorem is referenced by:  fsumf1o  15755  fprodf1o  15978  0ramcl  17056  fullthinc  48845
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