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Theorem fndmeng 6380
Description: A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
Assertion
Ref Expression
fndmeng  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  A  ~~  F )

Proof of Theorem fndmeng
StepHypRef Expression
1 fnex 5437 . . 3  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  F  e.  _V )
2 fnfun 5048 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
32adantr 270 . . 3  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  Fun  F )
4 fundmeng 6377 . . 3  |-  ( ( F  e.  _V  /\  Fun  F )  ->  dom  F 
~~  F )
51, 3, 4syl2anc 403 . 2  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  dom  F  ~~  F
)
6 fndm 5050 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
76breq1d 3816 . . 3  |-  ( F  Fn  A  ->  ( dom  F  ~~  F  <->  A  ~~  F ) )
87adantr 270 . 2  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  ( dom  F  ~~  F 
<->  A  ~~  F ) )
95, 8mpbid 145 1  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  A  ~~  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1434   _Vcvv 2611   class class class wbr 3806   dom cdm 4392   Fun wfun 4947    Fn wfn 4948    ~~ cen 6308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2613  df-sbc 2826  df-csb 2919  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-en 6311
This theorem is referenced by:  fihashfn  9901
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