Theorem fnfvima 5953

Description: The function value of an operand in a set is contained in the image of that set, using the Fn abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.)

Assertion

Ref	Expression
fnfvima	|- ( ( F Fn A /\ S C_ A /\ X e. S ) -> ( F ` X ) e. ( F " S ) )

Proof of Theorem fnfvima

Step	Hyp	Ref	Expression
1		fnfun 5478	. . . 4 |- ( F Fn A -> Fun F )
2	1	3ad2ant1 1049	. . 3 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> Fun F )
3		simp2 1029	. . . 4 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> S C_ A )
4		fndm 5480	. . . . 5 |- ( F Fn A -> dom F = A )
5	4	3ad2ant1 1049	. . . 4 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> dom F = A )
6	3, 5	sseqtrrd 3287	. . 3 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> S C_ dom F )
7	2, 6	jca 306	. 2 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> ( Fun F /\ S C_ dom F ) )
8		simp3 1030	. 2 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> X e. S )
9		funfvima2 5951	. 2 |- ( ( Fun F /\ S C_ dom F ) -> ( X e. S -> ( F ` X ) e. ( F " S ) ) )
10	7, 8, 9	sylc 62	1 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> ( F ` X ) e. ( F " S ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   dom cdm 4774   "cima 4777   Fun wfun 5371   Fn wfn 5372   ` cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385

This theorem is used by:  fnfvimad  5954  iseqf1olemnab  10938  ennnfonelemrn  13310  mhmima  13798  ghmnsgima  14071  lmtopcnp  15351