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Theorem fndmeng 6980
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 5782 . . 3  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  F  e.  _V )
2 fnfun 5378 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
32adantr 451 . . 3  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  Fun  F )
4 fundmeng 6978 . . 3  |-  ( ( F  e.  _V  /\  Fun  F )  ->  dom  F 
~~  F )
51, 3, 4syl2anc 642 . 2  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  dom  F  ~~  F
)
6 fndm 5380 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
76breq1d 4070 . . 3  |-  ( F  Fn  A  ->  ( dom  F  ~~  F  <->  A  ~~  F ) )
87adantr 451 . 2  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  ( dom  F  ~~  F 
<->  A  ~~  F ) )
95, 8mpbid 201 1  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  A  ~~  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1701   _Vcvv 2822   class class class wbr 4060   dom cdm 4726   Fun wfun 5286    Fn wfn 5287    ~~ cen 6903
This theorem is referenced by:  tskcard  8448  hashfn  11404  eupai  24167
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-en 6907
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