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Theorem funfvima2 6165
Description: A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.)
Assertion
Ref Expression
funfvima2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( B  e.  A  ->  ( F `  B
)  e.  ( F
" A ) ) )

Proof of Theorem funfvima2
StepHypRef Expression
1 ssel 3437 . . 3  |-  ( A 
C_  dom  F  ->  ( B  e.  A  ->  B  e.  dom  F ) )
2 funfvima 6164 . . . . . 6  |-  ( ( Fun  F  /\  B  e.  dom  F )  -> 
( B  e.  A  ->  ( F `  B
)  e.  ( F
" A ) ) )
32ex 440 . . . . 5  |-  ( Fun 
F  ->  ( B  e.  dom  F  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) )
43com23 81 . . . 4  |-  ( Fun 
F  ->  ( B  e.  A  ->  ( B  e.  dom  F  -> 
( F `  B
)  e.  ( F
" A ) ) ) )
54a2d 29 . . 3  |-  ( Fun 
F  ->  ( ( B  e.  A  ->  B  e.  dom  F )  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A ) ) ) )
61, 5syl5 33 . 2  |-  ( Fun 
F  ->  ( A  C_ 
dom  F  ->  ( B  e.  A  ->  ( F `  B )  e.  ( F " A
) ) ) )
76imp 435 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( B  e.  A  ->  ( F `  B
)  e.  ( F
" A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    e. wcel 1897    C_ wss 3415   dom cdm 4852   "cima 4855   Fun wfun 5594   ` cfv 5600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-fv 5608
This theorem is referenced by:  fnfvima  6167  f1oweALT  6803  tz7.49  7187  phimullem  14775  mrcuni  15575  frlmsslsp  19402  lindfrn  19427  iscldtop  20159  1stcfb  20508  2ndcomap  20521  rnelfm  21016  fmfnfmlem2  21018  fmfnfmlem4  21020  qtopbaslem  21827  tgqioo  21866  bndth  22034  volsup  22557  dyadmbllem  22605  opnmbllem  22607  itg1addlem4  22705  c1liplem1  22996  dvcnvrelem1  23017  dvcnvrelem2  23018  plyco0  23194  plyaddlem1  23215  plymullem1  23216  dvloglem  23641  logf1o2  23643  efopn  23651  axcontlem10  25051  eupares  25751  imaelshi  27759  funimass4f  28282  sitgclg  29223  cvmliftlem3  30058  nocvxminlem  30627  nocvxmin  30628  ivthALT  31039  opnmbllem0  32020  ismtyres  32184  heibor1lem  32185  ismrc  35587  aomclem4  35959  funfvima2d  36656
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