Theorem fnfvima 5797

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 5355	. . . 4 |- ( F Fn A -> Fun F )
2	1	3ad2ant1 1020	. . 3 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> Fun F )
3		simp2 1000	. . . 4 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> S C_ A )
4		fndm 5357	. . . . 5 |- ( F Fn A -> dom F = A )
5	4	3ad2ant1 1020	. . . 4 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> dom F = A )
6	3, 5	sseqtrrd 3222	. . 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 1001	. 2 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> X e. S )
9		funfvima2 5795	. 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 980   = wceq 1364   e. wcel 2167   C_ wss 3157   dom cdm 4663   "cima 4666   Fun wfun 5252   Fn wfn 5253   ` cfv 5258

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266

This theorem is referenced by:  iseqf1olemnab  10593  ennnfonelemrn  12636  mhmima  13123  ghmnsgima  13398  lmtopcnp  14486