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Theorem ndmfv 5714
Description: The value of a class outside its domain is the empty set. (Contributed by NM, 24-Aug-1995.)
Assertion
Ref Expression
ndmfv  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )

Proof of Theorem ndmfv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 euex 2277 . . . . 5  |-  ( E! x  A F x  ->  E. x  A F x )
2 eldmg 5024 . . . . 5  |-  ( A  e.  _V  ->  ( A  e.  dom  F  <->  E. x  A F x ) )
31, 2syl5ibr 213 . . . 4  |-  ( A  e.  _V  ->  ( E! x  A F x  ->  A  e.  dom  F ) )
43con3d 127 . . 3  |-  ( A  e.  _V  ->  ( -.  A  e.  dom  F  ->  -.  E! x  A F x ) )
5 tz6.12-2 5678 . . 3  |-  ( -.  E! x  A F x  ->  ( F `  A )  =  (/) )
64, 5syl6 31 . 2  |-  ( A  e.  _V  ->  ( -.  A  e.  dom  F  ->  ( F `  A )  =  (/) ) )
7 fvprc 5681 . . 3  |-  ( -.  A  e.  _V  ->  ( F `  A )  =  (/) )
87a1d 23 . 2  |-  ( -.  A  e.  _V  ->  ( -.  A  e.  dom  F  ->  ( F `  A )  =  (/) ) )
96, 8pm2.61i 158 1  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   E.wex 1547    = wceq 1649    e. wcel 1721   E!weu 2254   _Vcvv 2916   (/)c0 3588   class class class wbr 4172   dom cdm 4837   ` cfv 5413
This theorem is referenced by:  ndmfvrcl  5715  elfvdm  5716  nfvres  5719  fvfundmfvn0  5721  fv01  5722  funfv  5749  fvun1  5753  fvco4i  5760  fvmpti  5764  fvmptss  5772  fvmptex  5774  fvmptnf  5781  fvmptss2  5783  fvopab4ndm  5784  f0cli  5839  funiunfv  5954  ovprc  6067  oprssdm  6187  nssdmovg  6188  ndmovg  6189  1st2val  6331  2nd2val  6332  bropopvvv  6385  curry1val  6398  curry2val  6402  smofvon2  6577  rdgsucmptnf  6646  frsucmptn  6655  brwitnlem  6710  undifixp  7057  r1tr  7658  rankvaln  7681  cardidm  7802  carden2a  7809  carden2b  7810  carddomi2  7813  sdomsdomcardi  7814  pm54.43lem  7842  alephcard  7907  alephnbtwn  7908  alephgeom  7919  cfub  8085  cardcf  8088  cflecard  8089  cfle  8090  cflim2  8099  cfidm  8111  itunisuc  8255  itunitc1  8256  ituniiun  8258  alephadd  8408  alephreg  8413  pwcfsdom  8414  cfpwsdom  8415  adderpq  8789  mulerpq  8790  uzssz  10461  ltweuz  11256  swrd00  11720  sumz  12471  sumss  12473  sumnul  12499  divsfval  13727  cidpropd  13891  homarcl  14138  arwval  14153  coafval  14174  iscnp2  17257  setsmstopn  18461  tngtopn  18644  pcofval  18988  dvbsss  19742  perfdvf  19743  mpfrcl  19892  dchrrcl  20977  vsfval  22067  dmadjrnb  23362  hmdmadj  23396  gsumpropd2lem  24173  prod1  25223  prodss  25226  rdgprc0  25364  soseq  25468  nofv  25525  sltres  25532  bdayelon  25548  fvnobday  25550  fullfunfv  25700  eleenn  25739  linedegen  25981  itgocn  27237  2wlkonot3v  28072  2spthonot3v  28073  dibvalrel  31646  dicvalrelN  31668  dihvalrel  31762
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-nul 4298  ax-pow 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-dm 4847  df-iota 5377  df-fv 5421
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