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Theorem fundmeng 6287
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 4921 . . . 4  |-  ( x  =  F  ->  ( Fun  x  <->  Fun  F ) )
2 dmeq 4535 . . . . 5  |-  ( x  =  F  ->  dom  x  =  dom  F )
3 id 19 . . . . 5  |-  ( x  =  F  ->  x  =  F )
42, 3breq12d 3777 . . . 4  |-  ( x  =  F  ->  ( dom  x  ~~  x  <->  dom  F  ~~  F ) )
51, 4imbi12d 223 . . 3  |-  ( x  =  F  ->  (
( Fun  x  ->  dom  x  ~~  x )  <-> 
( Fun  F  ->  dom 
F  ~~  F )
) )
6 vex 2560 . . . 4  |-  x  e. 
_V
76fundmen 6286 . . 3  |-  ( Fun  x  ->  dom  x  ~~  x )
85, 7vtoclg 2613 . 2  |-  ( F  e.  V  ->  ( Fun  F  ->  dom  F  ~~  F ) )
98imp 115 1  |-  ( ( F  e.  V  /\  Fun  F )  ->  dom  F 
~~  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   class class class wbr 3764   dom cdm 4345   Fun wfun 4896    ~~ cen 6219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-en 6222
This theorem is referenced by:  fndmeng  6289
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