Theorem funfvima2 5872

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 3218	. . 3 |- ( A C_ dom F -> ( B e. A -> B e. dom F ) )
2		funfvima 5871	. . . . . 6 |- ( ( Fun F /\ B e. dom F ) -> ( B e. A -> ( F ` B ) e. ( F " A ) ) )
3	2	ex 115	. . . . 5 |- ( Fun F -> ( B e. dom F -> ( B e. A -> ( F ` B ) e. ( F " A ) ) ) )
4	3	com23 78	. . . 4 |- ( Fun F -> ( B e. A -> ( B e. dom F -> ( F ` B ) e. ( F " A ) ) ) )
5	4	a2d 26	. . 3 |- ( Fun F -> ( ( B e. A -> B e. dom F ) -> ( B e. A -> ( F ` B ) e. ( F " A ) ) ) )
6	1, 5	syl5 32	. 2 |- ( Fun F -> ( A C_ dom F -> ( B e. A -> ( F ` B ) e. ( F " A ) ) ) )
7	6	imp 124	1 |- ( ( Fun F /\ A C_ dom F ) -> ( B e. A -> ( F ` B ) e. ( F " A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   C_ wss 3197   dom cdm 4719   "cima 4722   Fun wfun 5312   ` cfv 5318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326

This theorem is referenced by:  fnfvima  5874  phimullem  12747  qtopbasss  15195  tgqioo  15229  plyaddlem1  15421  plymullem1  15422