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Theorem fndmeng 6704
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 5642 . . 3  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  F  e.  _V )
2 fnfun 5220 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
32adantr 274 . . 3  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  Fun  F )
4 fundmeng 6701 . . 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 5222 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
76breq1d 3939 . . 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 1480   _Vcvv 2686   class class class wbr 3929   dom cdm 4539   Fun wfun 5117    Fn wfn 5118    ~~ cen 6632
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-en 6635
This theorem is referenced by:  fihashfn  10553
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