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Theorem fo00 6391
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 6333 . . . . . 6 (𝐹:∅–onto𝐴𝐹 Fn ∅)
2 fn0 6222 . . . . . . 7 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
3 f10 6388 . . . . . . . 8 ∅:∅–1-1𝐴
4 f1eq1 6311 . . . . . . . 8 (𝐹 = ∅ → (𝐹:∅–1-1𝐴 ↔ ∅:∅–1-1𝐴))
53, 4mpbiri 250 . . . . . . 7 (𝐹 = ∅ → 𝐹:∅–1-1𝐴)
62, 5sylbi 209 . . . . . 6 (𝐹 Fn ∅ → 𝐹:∅–1-1𝐴)
71, 6syl 17 . . . . 5 (𝐹:∅–onto𝐴𝐹:∅–1-1𝐴)
87ancri 546 . . . 4 (𝐹:∅–onto𝐴 → (𝐹:∅–1-1𝐴𝐹:∅–onto𝐴))
9 df-f1o 6108 . . . 4 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹:∅–1-1𝐴𝐹:∅–onto𝐴))
108, 9sylibr 226 . . 3 (𝐹:∅–onto𝐴𝐹:∅–1-1-onto𝐴)
11 f1ofo 6363 . . 3 (𝐹:∅–1-1-onto𝐴𝐹:∅–onto𝐴)
1210, 11impbii 201 . 2 (𝐹:∅–onto𝐴𝐹:∅–1-1-onto𝐴)
13 f1o00 6390 . 2 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
1412, 13bitri 267 1 (𝐹:∅–onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385   = wceq 1653  c0 4115   Fn wfn 6096  1-1wf1 6098  ontowfo 6099  1-1-ontowf1o 6100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108
This theorem is referenced by:  fsumf1o  14795  fprodf1o  15013  0ramcl  16060
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