Theorem funfvima2 6493

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

Step	Hyp	Ref	Expression
1		ssel 3597	. . 3 |- ( A C_ dom F -> ( B e. A -> B e. dom F ) )
2		funfvima 6492	. . . . . 6 |- ( ( Fun F /\ B e. dom F ) -> ( B e. A -> ( F ` B ) e. ( F " A ) ) )
3	2	ex 450	. . . . 5 |- ( Fun F -> ( B e. dom F -> ( B e. A -> ( F ` B ) e. ( F " A ) ) ) )
4	3	com23 86	. . . 4 |- ( Fun F -> ( B e. A -> ( B e. dom F -> ( F ` B ) e. ( F " A ) ) ) )
5	4	a2d 29	. . 3 |- ( Fun F -> ( ( B e. A -> B e. dom F ) -> ( B e. A -> ( F ` B ) e. ( F " A ) ) ) )
6	1, 5	syl5 34	. 2 |- ( Fun F -> ( A C_ dom F -> ( B e. A -> ( F ` B ) e. ( F " A ) ) ) )
7	6	imp 445	1 |- ( ( Fun F /\ A C_ dom F ) -> ( B e. A -> ( F ` B ) e. ( F " A ) ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   C_ wss 3574   dom cdm 5114   "cima 5117   Fun wfun 5882   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  resfvresima  6494  fnfvima  6496  f1oweALT  7152  tz7.49  7540  phimullem  15484  mrcuni  16281  frlmsslsp  20135  lindfrn  20160  iscldtop  20899  1stcfb  21248  2ndcomap  21261  rnelfm  21757  fmfnfmlem2  21759  fmfnfmlem4  21761  qtopbaslem  22562  tgqioo  22603  bndth  22757  volsup  23324  dyadmbllem  23367  opnmbllem  23369  itg1addlem4  23466  c1liplem1  23759  dvcnvrelem1  23780  dvcnvrelem2  23781  plyco0  23948  plyaddlem1  23969  plymullem1  23970  dvloglem  24394  logf1o2  24396  efopn  24404  axcontlem10  25853  imaelshi  28917  funimass4f  29437  sitgclg  30404  cvmliftlem3  31269  nocvxminlem  31893  nocvxmin  31894  ivthALT  32330  opnmbllem0  33445  ismtyres  33607  heibor1lem  33608  ismrc  37264  aomclem4  37627  funfvima2d  38469  fnfvimad  39459