Theorem fnfvima 5926

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 5458	. . . 4 |- ( F Fn A -> Fun F )
2	1	3ad2ant1 1045	. . 3 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> Fun F )
3		simp2 1025	. . . 4 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> S C_ A )
4		fndm 5460	. . . . 5 |- ( F Fn A -> dom F = A )
5	4	3ad2ant1 1045	. . . 4 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> dom F = A )
6	3, 5	sseqtrrd 3281	. . 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 1026	. 2 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> X e. S )
9		funfvima2 5924	. 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   C_ wss 3214   dom cdm 4754   "cima 4757   Fun wfun 5351   Fn wfn 5352   ` cfv 5357

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365

This theorem is referenced by:  fnfvimad  5927  iseqf1olemnab  10887  ennnfonelemrn  13254  mhmima  13746  ghmnsgima  14021  lmtopcnp  15241