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Theorem f1o00 5189
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 5162 . 2  |-  ( F : (/)
-1-1-onto-> A 
<->  ( F  Fn  (/)  /\  `' F  Fn  A )
)
2 fn0 5046 . . . . . 6  |-  ( F  Fn  (/)  <->  F  =  (/) )
32biimpi 117 . . . . 5  |-  ( F  Fn  (/)  ->  F  =  (/) )
43adantr 265 . . . 4  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  F  =  (/) )
5 dm0 4577 . . . . 5  |-  dom  (/)  =  (/)
6 cnveq 4537 . . . . . . . . . 10  |-  ( F  =  (/)  ->  `' F  =  `' (/) )
7 cnv0 4755 . . . . . . . . . 10  |-  `' (/)  =  (/)
86, 7syl6eq 2104 . . . . . . . . 9  |-  ( F  =  (/)  ->  `' F  =  (/) )
92, 8sylbi 118 . . . . . . . 8  |-  ( F  Fn  (/)  ->  `' F  =  (/) )
109fneq1d 5017 . . . . . . 7  |-  ( F  Fn  (/)  ->  ( `' F  Fn  A  <->  (/)  Fn  A
) )
1110biimpa 284 . . . . . 6  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  -> 
(/)  Fn  A )
12 fndm 5026 . . . . . 6  |-  ( (/)  Fn  A  ->  dom  (/)  =  A )
1311, 12syl 14 . . . . 5  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  dom  (/)  =  A )
145, 13syl5reqr 2103 . . . 4  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  A  =  (/) )
154, 14jca 294 . . 3  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  ( F  =  (/)  /\  A  =  (/) ) )
162biimpri 128 . . . . 5  |-  ( F  =  (/)  ->  F  Fn  (/) )
1716adantr 265 . . . 4  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  F  Fn  (/) )
18 eqid 2056 . . . . . 6  |-  (/)  =  (/)
19 fn0 5046 . . . . . 6  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
2018, 19mpbir 138 . . . . 5  |-  (/)  Fn  (/)
218fneq1d 5017 . . . . . 6  |-  ( F  =  (/)  ->  ( `' F  Fn  A  <->  (/)  Fn  A
) )
22 fneq2 5016 . . . . . 6  |-  ( A  =  (/)  ->  ( (/)  Fn  A  <->  (/)  Fn  (/) ) )
2321, 22sylan9bb 443 . . . . 5  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  ( `' F  Fn  A  <->  (/)  Fn  (/) ) )
2420, 23mpbiri 161 . . . 4  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  `' F  Fn  A )
2517, 24jca 294 . . 3  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  ( F  Fn  (/)  /\  `' F  Fn  A )
)
2615, 25impbii 121 . 2  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  <->  ( F  =  (/)  /\  A  =  (/) ) )
271, 26bitri 177 1  |-  ( F : (/)
-1-1-onto-> A 
<->  ( F  =  (/)  /\  A  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    = wceq 1259   (/)c0 3252   `'ccnv 4372   dom cdm 4373    Fn wfn 4925   -1-1-onto->wf1o 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937
This theorem is referenced by:  fo00  5190  f1o0  5191  en0  6306
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