MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fundmeng Unicode version

Theorem fundmeng 7172
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 5464 . . . 4  |-  ( x  =  F  ->  ( Fun  x  <->  Fun  F ) )
2 dmeq 5061 . . . . 5  |-  ( x  =  F  ->  dom  x  =  dom  F )
3 id 20 . . . . 5  |-  ( x  =  F  ->  x  =  F )
42, 3breq12d 4217 . . . 4  |-  ( x  =  F  ->  ( dom  x  ~~  x  <->  dom  F  ~~  F ) )
51, 4imbi12d 312 . . 3  |-  ( x  =  F  ->  (
( Fun  x  ->  dom  x  ~~  x )  <-> 
( Fun  F  ->  dom 
F  ~~  F )
) )
6 vex 2951 . . . 4  |-  x  e. 
_V
76fundmen 7171 . . 3  |-  ( Fun  x  ->  dom  x  ~~  x )
85, 7vtoclg 3003 . 2  |-  ( F  e.  V  ->  ( Fun  F  ->  dom  F  ~~  F ) )
98imp 419 1  |-  ( ( F  e.  V  /\  Fun  F )  ->  dom  F 
~~  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4204   dom cdm 4869   Fun wfun 5439    ~~ cen 7097
This theorem is referenced by:  fndmeng  7174
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-en 7101
  Copyright terms: Public domain W3C validator