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Theorem fundmeng 7590
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 5607 . . . 4  |-  ( x  =  F  ->  ( Fun  x  <->  Fun  F ) )
2 dmeq 5203 . . . . 5  |-  ( x  =  F  ->  dom  x  =  dom  F )
3 id 22 . . . . 5  |-  ( x  =  F  ->  x  =  F )
42, 3breq12d 4460 . . . 4  |-  ( x  =  F  ->  ( dom  x  ~~  x  <->  dom  F  ~~  F ) )
51, 4imbi12d 320 . . 3  |-  ( x  =  F  ->  (
( Fun  x  ->  dom  x  ~~  x )  <-> 
( Fun  F  ->  dom 
F  ~~  F )
) )
6 vex 3116 . . . 4  |-  x  e. 
_V
76fundmen 7589 . . 3  |-  ( Fun  x  ->  dom  x  ~~  x )
85, 7vtoclg 3171 . 2  |-  ( F  e.  V  ->  ( Fun  F  ->  dom  F  ~~  F ) )
98imp 429 1  |-  ( ( F  e.  V  /\  Fun  F )  ->  dom  F 
~~  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   class class class wbr 4447   dom cdm 4999   Fun wfun 5582    ~~ cen 7513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-en 7517
This theorem is referenced by:  fndmeng  7592
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