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Theorem f1o00 5830
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 5806 . 2  |-  ( F : (/)
-1-1-onto-> A 
<->  ( F  Fn  (/)  /\  `' F  Fn  A )
)
2 fn0 5682 . . . . . 6  |-  ( F  Fn  (/)  <->  F  =  (/) )
32biimpi 194 . . . . 5  |-  ( F  Fn  (/)  ->  F  =  (/) )
43adantr 463 . . . 4  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  F  =  (/) )
5 dm0 5205 . . . . 5  |-  dom  (/)  =  (/)
6 cnveq 5165 . . . . . . . . . 10  |-  ( F  =  (/)  ->  `' F  =  `' (/) )
7 cnv0 5394 . . . . . . . . . 10  |-  `' (/)  =  (/)
86, 7syl6eq 2511 . . . . . . . . 9  |-  ( F  =  (/)  ->  `' F  =  (/) )
92, 8sylbi 195 . . . . . . . 8  |-  ( F  Fn  (/)  ->  `' F  =  (/) )
109fneq1d 5653 . . . . . . 7  |-  ( F  Fn  (/)  ->  ( `' F  Fn  A  <->  (/)  Fn  A
) )
1110biimpa 482 . . . . . 6  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  -> 
(/)  Fn  A )
12 fndm 5662 . . . . . 6  |-  ( (/)  Fn  A  ->  dom  (/)  =  A )
1311, 12syl 16 . . . . 5  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  dom  (/)  =  A )
145, 13syl5reqr 2510 . . . 4  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  A  =  (/) )
154, 14jca 530 . . 3  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  ( F  =  (/)  /\  A  =  (/) ) )
162biimpri 206 . . . . 5  |-  ( F  =  (/)  ->  F  Fn  (/) )
1716adantr 463 . . . 4  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  F  Fn  (/) )
18 eqid 2454 . . . . . 6  |-  (/)  =  (/)
19 fn0 5682 . . . . . 6  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
2018, 19mpbir 209 . . . . 5  |-  (/)  Fn  (/)
218fneq1d 5653 . . . . . 6  |-  ( F  =  (/)  ->  ( `' F  Fn  A  <->  (/)  Fn  A
) )
22 fneq2 5652 . . . . . 6  |-  ( A  =  (/)  ->  ( (/)  Fn  A  <->  (/)  Fn  (/) ) )
2321, 22sylan9bb 697 . . . . 5  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  ( `' F  Fn  A  <->  (/)  Fn  (/) ) )
2420, 23mpbiri 233 . . . 4  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  `' F  Fn  A )
2517, 24jca 530 . . 3  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  ( F  Fn  (/)  /\  `' F  Fn  A )
)
2615, 25impbii 188 . 2  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  <->  ( F  =  (/)  /\  A  =  (/) ) )
271, 26bitri 249 1  |-  ( F : (/)
-1-1-onto-> A 
<->  ( F  =  (/)  /\  A  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1398   (/)c0 3783   `'ccnv 4987   dom cdm 4988    Fn wfn 5565   -1-1-onto->wf1o 5569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577
This theorem is referenced by:  fo00  5831  f1o0  5832  en0  7571  symgbas0  16621  derang0  28880
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