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Theorem fndmeng 7582
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 6118 . . 3  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  F  e.  _V )
2 fnfun 5669 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
32adantr 465 . . 3  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  Fun  F )
4 fundmeng 7580 . . 3  |-  ( ( F  e.  _V  /\  Fun  F )  ->  dom  F 
~~  F )
51, 3, 4syl2anc 661 . 2  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  dom  F  ~~  F
)
6 fndm 5671 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
76breq1d 4450 . . 3  |-  ( F  Fn  A  ->  ( dom  F  ~~  F  <->  A  ~~  F ) )
87adantr 465 . 2  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  ( dom  F  ~~  F 
<->  A  ~~  F ) )
95, 8mpbid 210 1  |-  ( ( F  Fn  A  /\  A  e.  C )  ->  A  ~~  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1762   _Vcvv 3106   class class class wbr 4440   dom cdm 4992   Fun wfun 5573    Fn wfn 5574    ~~ cen 7503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-en 7507
This theorem is referenced by:  tskcard  9148  hashfn  12398  eupai  24629
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