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Theorem fundmeng 7651
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 )

Proof of Theorem fundmeng
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 funeq 5620 . . . 4  |-  ( x  =  F  ->  ( Fun  x  <->  Fun  F ) )
2 dmeq 5055 . . . . 5  |-  ( x  =  F  ->  dom  x  =  dom  F )
3 id 23 . . . . 5  |-  ( x  =  F  ->  x  =  F )
42, 3breq12d 4439 . . . 4  |-  ( x  =  F  ->  ( dom  x  ~~  x  <->  dom  F  ~~  F ) )
51, 4imbi12d 321 . . 3  |-  ( x  =  F  ->  (
( Fun  x  ->  dom  x  ~~  x )  <-> 
( Fun  F  ->  dom 
F  ~~  F )
) )
6 vex 3090 . . . 4  |-  x  e. 
_V
76fundmen 7650 . . 3  |-  ( Fun  x  ->  dom  x  ~~  x )
85, 7vtoclg 3145 . 2  |-  ( F  e.  V  ->  ( Fun  F  ->  dom  F  ~~  F ) )
98imp 430 1  |-  ( ( F  e.  V  /\  Fun  F )  ->  dom  F 
~~  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   class class class wbr 4426   dom cdm 4854   Fun wfun 5595    ~~ cen 7574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-int 4259  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-en 7578
This theorem is referenced by:  fndmeng  7653
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