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Theorem fo00 6314
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 6259 . . . . . 6 (𝐹:∅–onto𝐴𝐹 Fn ∅)
2 fn0 6150 . . . . . . 7 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
3 f10 6311 . . . . . . . 8 ∅:∅–1-1𝐴
4 f1eq1 6237 . . . . . . . 8 (𝐹 = ∅ → (𝐹:∅–1-1𝐴 ↔ ∅:∅–1-1𝐴))
53, 4mpbiri 248 . . . . . . 7 (𝐹 = ∅ → 𝐹:∅–1-1𝐴)
62, 5sylbi 207 . . . . . 6 (𝐹 Fn ∅ → 𝐹:∅–1-1𝐴)
71, 6syl 17 . . . . 5 (𝐹:∅–onto𝐴𝐹:∅–1-1𝐴)
87ancri 539 . . . 4 (𝐹:∅–onto𝐴 → (𝐹:∅–1-1𝐴𝐹:∅–onto𝐴))
9 df-f1o 6037 . . . 4 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹:∅–1-1𝐴𝐹:∅–onto𝐴))
108, 9sylibr 224 . . 3 (𝐹:∅–onto𝐴𝐹:∅–1-1-onto𝐴)
11 f1ofo 6286 . . 3 (𝐹:∅–1-1-onto𝐴𝐹:∅–onto𝐴)
1210, 11impbii 199 . 2 (𝐹:∅–onto𝐴𝐹:∅–1-1-onto𝐴)
13 f1o00 6313 . 2 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
1412, 13bitri 264 1 (𝐹:∅–onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  c0 4063   Fn wfn 6025  1-1wf1 6027  ontowfo 6028  1-1-ontowf1o 6029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037
This theorem is referenced by:  fsumf1o  14662  fprodf1o  14883  0ramcl  15934
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