Theorem fnfvima 5873

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 5417	. . . 4 |- ( F Fn A -> Fun F )
2	1	3ad2ant1 1042	. . 3 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> Fun F )
3		simp2 1022	. . . 4 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> S C_ A )
4		fndm 5419	. . . . 5 |- ( F Fn A -> dom F = A )
5	4	3ad2ant1 1042	. . . 4 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> dom F = A )
6	3, 5	sseqtrrd 3263	. . 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 1023	. 2 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> X e. S )
9		funfvima2 5871	. 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 1002   = wceq 1395   e. wcel 2200   C_ wss 3197   dom cdm 4718   "cima 4721   Fun wfun 5311   Fn wfn 5312   ` cfv 5317

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 4201  ax-pow 4257  ax-pr 4292

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 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325

This theorem is referenced by:  iseqf1olemnab  10718  ennnfonelemrn  12985  mhmima  13519  ghmnsgima  13800  lmtopcnp  14918