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Theorem fundmeng 6931
Description: A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 17-Sep-2013.)
Assertion
Ref Expression
fundmeng  |-  ( ( F  e.  V  /\  Fun  F )  ->  dom  F 
~~  F )
Dummy variable  x is distinct from all other variables.

Proof of Theorem fundmeng
StepHypRef Expression
1 funeq 5241 . . . 4  |-  ( x  =  F  ->  ( Fun  x  <->  Fun  F ) )
2 dmeq 4879 . . . . 5  |-  ( x  =  F  ->  dom  x  =  dom  F )
3 id 21 . . . . 5  |-  ( x  =  F  ->  x  =  F )
42, 3breq12d 4038 . . . 4  |-  ( x  =  F  ->  ( dom  x  ~~  x  <->  dom  F  ~~  F ) )
51, 4imbi12d 313 . . 3  |-  ( x  =  F  ->  (
( Fun  x  ->  dom  x  ~~  x )  <-> 
( Fun  F  ->  dom 
F  ~~  F )
) )
6 vex 2793 . . . 4  |-  x  e. 
_V
76fundmen 6930 . . 3  |-  ( Fun  x  ->  dom  x  ~~  x )
85, 7vtoclg 2845 . 2  |-  ( F  e.  V  ->  ( Fun  F  ->  dom  F  ~~  F ) )
98imp 420 1  |-  ( ( F  e.  V  /\  Fun  F )  ->  dom  F 
~~  F )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685   class class class wbr 4025   dom cdm 4689   Fun wfun 5216    ~~ cen 6856
This theorem is referenced by:  fndmeng  6933
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-int 3865  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-en 6860
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