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Theorem fndmeng 6672
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 5610 . . 3  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  F  e.  _V )
2 fnfun 5190 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
32adantr 274 . . 3  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  Fun  F )
4 fundmeng 6669 . . 3  |-  ( ( F  e.  _V  /\  Fun  F )  ->  dom  F 
~~  F )
51, 3, 4syl2anc 408 . 2  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  dom  F  ~~  F
)
6 fndm 5192 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
76breq1d 3909 . . 3  |-  ( F  Fn  A  ->  ( dom  F  ~~  F  <->  A  ~~  F ) )
87adantr 274 . 2  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  ( dom  F  ~~  F 
<->  A  ~~  F ) )
95, 8mpbid 146 1  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  A  ~~  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1465   _Vcvv 2660   class class class wbr 3899   dom cdm 4509   Fun wfun 5087    Fn wfn 5088    ~~ cen 6600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-en 6603
This theorem is referenced by:  fihashfn  10514
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