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Theorem f1o00 5776
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 5752 . 2  |-  ( F : (/)
-1-1-onto-> A 
<->  ( F  Fn  (/)  /\  `' F  Fn  A )
)
2 fn0 5633 . . . . . 6  |-  ( F  Fn  (/)  <->  F  =  (/) )
32biimpi 194 . . . . 5  |-  ( F  Fn  (/)  ->  F  =  (/) )
43adantr 465 . . . 4  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  F  =  (/) )
5 dm0 5156 . . . . 5  |-  dom  (/)  =  (/)
6 cnveq 5116 . . . . . . . . . 10  |-  ( F  =  (/)  ->  `' F  =  `' (/) )
7 cnv0 5343 . . . . . . . . . 10  |-  `' (/)  =  (/)
86, 7syl6eq 2509 . . . . . . . . 9  |-  ( F  =  (/)  ->  `' F  =  (/) )
92, 8sylbi 195 . . . . . . . 8  |-  ( F  Fn  (/)  ->  `' F  =  (/) )
109fneq1d 5604 . . . . . . 7  |-  ( F  Fn  (/)  ->  ( `' F  Fn  A  <->  (/)  Fn  A
) )
1110biimpa 484 . . . . . 6  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  -> 
(/)  Fn  A )
12 fndm 5613 . . . . . 6  |-  ( (/)  Fn  A  ->  dom  (/)  =  A )
1311, 12syl 16 . . . . 5  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  dom  (/)  =  A )
145, 13syl5reqr 2508 . . . 4  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  A  =  (/) )
154, 14jca 532 . . 3  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  ( F  =  (/)  /\  A  =  (/) ) )
162biimpri 206 . . . . 5  |-  ( F  =  (/)  ->  F  Fn  (/) )
1716adantr 465 . . . 4  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  F  Fn  (/) )
18 eqid 2452 . . . . . 6  |-  (/)  =  (/)
19 fn0 5633 . . . . . 6  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
2018, 19mpbir 209 . . . . 5  |-  (/)  Fn  (/)
218fneq1d 5604 . . . . . 6  |-  ( F  =  (/)  ->  ( `' F  Fn  A  <->  (/)  Fn  A
) )
22 fneq2 5603 . . . . . 6  |-  ( A  =  (/)  ->  ( (/)  Fn  A  <->  (/)  Fn  (/) ) )
2321, 22sylan9bb 699 . . . . 5  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  ( `' F  Fn  A  <->  (/)  Fn  (/) ) )
2420, 23mpbiri 233 . . . 4  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  `' F  Fn  A )
2517, 24jca 532 . . 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 369    = wceq 1370   (/)c0 3740   `'ccnv 4942   dom cdm 4943    Fn wfn 5516   -1-1-onto->wf1o 5520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-br 4396  df-opab 4454  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528
This theorem is referenced by:  fo00  5777  f1o0  5778  en0  7477  symgbas0  16013  derang0  27196
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