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Theorem fo00 6632
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 6574 . . . . . 6 (𝐹:∅–onto𝐴𝐹 Fn ∅)
2 fn0 6459 . . . . . . 7 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
3 f10 6629 . . . . . . . 8 ∅:∅–1-1𝐴
4 f1eq1 6551 . . . . . . . 8 (𝐹 = ∅ → (𝐹:∅–1-1𝐴 ↔ ∅:∅–1-1𝐴))
53, 4mpbiri 261 . . . . . . 7 (𝐹 = ∅ → 𝐹:∅–1-1𝐴)
62, 5sylbi 220 . . . . . 6 (𝐹 Fn ∅ → 𝐹:∅–1-1𝐴)
71, 6syl 17 . . . . 5 (𝐹:∅–onto𝐴𝐹:∅–1-1𝐴)
87ancri 553 . . . 4 (𝐹:∅–onto𝐴 → (𝐹:∅–1-1𝐴𝐹:∅–onto𝐴))
9 df-f1o 6341 . . . 4 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹:∅–1-1𝐴𝐹:∅–onto𝐴))
108, 9sylibr 237 . . 3 (𝐹:∅–onto𝐴𝐹:∅–1-1-onto𝐴)
11 f1ofo 6604 . . 3 (𝐹:∅–1-1-onto𝐴𝐹:∅–onto𝐴)
1210, 11impbii 212 . 2 (𝐹:∅–onto𝐴𝐹:∅–1-1-onto𝐴)
13 f1o00 6631 . 2 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
1412, 13bitri 278 1 (𝐹:∅–onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  c0 4265   Fn wfn 6329  1-1wf1 6331  ontowfo 6332  1-1-ontowf1o 6333
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341
This theorem is referenced by:  fsumf1o  15071  fprodf1o  15291  0ramcl  16348
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