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Theorem f1o00 5669
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 5641 . 2  |-  ( F : (/)
-1-1-onto-> A 
<->  ( F  Fn  (/)  /\  `' F  Fn  A )
)
2 fn0 5523 . . . . . 6  |-  ( F  Fn  (/)  <->  F  =  (/) )
32biimpi 187 . . . . 5  |-  ( F  Fn  (/)  ->  F  =  (/) )
43adantr 452 . . . 4  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  F  =  (/) )
5 dm0 5042 . . . . 5  |-  dom  (/)  =  (/)
6 cnveq 5005 . . . . . . . . . 10  |-  ( F  =  (/)  ->  `' F  =  `' (/) )
7 cnv0 5234 . . . . . . . . . 10  |-  `' (/)  =  (/)
86, 7syl6eq 2452 . . . . . . . . 9  |-  ( F  =  (/)  ->  `' F  =  (/) )
92, 8sylbi 188 . . . . . . . 8  |-  ( F  Fn  (/)  ->  `' F  =  (/) )
109fneq1d 5495 . . . . . . 7  |-  ( F  Fn  (/)  ->  ( `' F  Fn  A  <->  (/)  Fn  A
) )
1110biimpa 471 . . . . . 6  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  -> 
(/)  Fn  A )
12 fndm 5503 . . . . . 6  |-  ( (/)  Fn  A  ->  dom  (/)  =  A )
1311, 12syl 16 . . . . 5  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  dom  (/)  =  A )
145, 13syl5reqr 2451 . . . 4  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  A  =  (/) )
154, 14jca 519 . . 3  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  ->  ( F  =  (/)  /\  A  =  (/) ) )
162biimpri 198 . . . . 5  |-  ( F  =  (/)  ->  F  Fn  (/) )
1716adantr 452 . . . 4  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  F  Fn  (/) )
18 eqid 2404 . . . . . 6  |-  (/)  =  (/)
19 fn0 5523 . . . . . 6  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
2018, 19mpbir 201 . . . . 5  |-  (/)  Fn  (/)
218fneq1d 5495 . . . . . 6  |-  ( F  =  (/)  ->  ( `' F  Fn  A  <->  (/)  Fn  A
) )
22 fneq2 5494 . . . . . 6  |-  ( A  =  (/)  ->  ( (/)  Fn  A  <->  (/)  Fn  (/) ) )
2321, 22sylan9bb 681 . . . . 5  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  ( `' F  Fn  A  <->  (/)  Fn  (/) ) )
2420, 23mpbiri 225 . . . 4  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  `' F  Fn  A )
2517, 24jca 519 . . 3  |-  ( ( F  =  (/)  /\  A  =  (/) )  ->  ( F  Fn  (/)  /\  `' F  Fn  A )
)
2615, 25impbii 181 . 2  |-  ( ( F  Fn  (/)  /\  `' F  Fn  A )  <->  ( F  =  (/)  /\  A  =  (/) ) )
271, 26bitri 241 1  |-  ( F : (/)
-1-1-onto-> A 
<->  ( F  =  (/)  /\  A  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649   (/)c0 3588   `'ccnv 4836   dom cdm 4837    Fn wfn 5408   -1-1-onto->wf1o 5412
This theorem is referenced by:  fo00  5670  f1o0  5671  en0  7129  derang0  24808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420
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