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Theorem f1o00 5515
Description: One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
Assertion
Ref Expression
f1o00  |-  ( F : (/)
-1-1-onto-> A 
<->  ( F  =  (/)  /\  A  =  (/) ) )

Proof of Theorem f1o00
StepHypRef Expression
1 dff1o4 5488 . 2  |-  ( F : (/)
-1-1-onto-> A 
<->  ( F  Fn  (/)  /\  `' F  Fn  A )
)
2 fn0 5354 . . . . . 6  |-  ( F  Fn  (/)  <->  F  =  (/) )
32biimpi 120 . . . . 5  |-  ( F  Fn  (/)  ->  F  =  (/) )
43adantr 276 . . . 4  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  F  =  (/) )
5 cnveq 4819 . . . . . . . . . 10  |-  ( F  =  (/)  ->  `' F  =  `' (/) )
6 cnv0 5050 . . . . . . . . . 10  |-  `' (/)  =  (/)
75, 6eqtrdi 2238 . . . . . . . . 9  |-  ( F  =  (/)  ->  `' F  =  (/) )
82, 7sylbi 121 . . . . . . . 8  |-  ( F  Fn  (/)  ->  `' F  =  (/) )
98fneq1d 5325 . . . . . . 7  |-  ( F  Fn  (/)  ->  ( `' F  Fn  A  <->  (/)  Fn  A
) )
109biimpa 296 . . . . . 6  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  -> 
(/)  Fn  A )
11 fndm 5334 . . . . . 6  |-  ( (/)  Fn  A  ->  dom  (/)  =  A )
1210, 11syl 14 . . . . 5  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  dom  (/)  =  A )
13 dm0 4859 . . . . 5  |-  dom  (/)  =  (/)
1412, 13eqtr3di 2237 . . . 4  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  A  =  (/) )
154, 14jca 306 . . 3  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  ( F  =  (/)  /\  A  =  (/) ) )
162biimpri 133 . . . . 5  |-  ( F  =  (/)  ->  F  Fn  (/) )
1716adantr 276 . . . 4  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  F  Fn  (/) )
18 eqid 2189 . . . . . 6  |-  (/)  =  (/)
19 fn0 5354 . . . . . 6  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
2018, 19mpbir 146 . . . . 5  |-  (/)  Fn  (/)
217fneq1d 5325 . . . . . 6  |-  ( F  =  (/)  ->  ( `' F  Fn  A  <->  (/)  Fn  A
) )
22 fneq2 5324 . . . . . 6  |-  ( A  =  (/)  ->  ( (/)  Fn  A  <->  (/)  Fn  (/) ) )
2321, 22sylan9bb 462 . . . . 5  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  ( `' F  Fn  A  <->  (/)  Fn  (/) ) )
2420, 23mpbiri 168 . . . 4  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  `' F  Fn  A )
2517, 24jca 306 . . 3  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  ( F  Fn  (/)  /\  `' F  Fn  A )
)
2615, 25impbii 126 . 2  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  <->  ( F  =  (/)  /\  A  =  (/) ) )
271, 26bitri 184 1  |-  ( F : (/)
-1-1-onto-> A 
<->  ( F  =  (/)  /\  A  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1364   (/)c0 3437   `'ccnv 4643   dom cdm 4644    Fn wfn 5230   -1-1-onto->wf1o 5234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242
This theorem is referenced by:  fo00  5516  f1o0  5517  en0  6820
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