Theorem fnfvima 5837

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 5385	. . . 4 |- ( F Fn A -> Fun F )
2	1	3ad2ant1 1021	. . 3 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> Fun F )
3		simp2 1001	. . . 4 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> S C_ A )
4		fndm 5387	. . . . 5 |- ( F Fn A -> dom F = A )
5	4	3ad2ant1 1021	. . . 4 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> dom F = A )
6	3, 5	sseqtrrd 3236	. . 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 1002	. 2 |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> X e. S )
9		funfvima2 5835	. 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 981   = wceq 1373   e. wcel 2177   C_ wss 3170   dom cdm 4688   "cima 4691   Fun wfun 5279   Fn wfn 5280   ` cfv 5285

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-opab 4117  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-fv 5293

This theorem is referenced by:  iseqf1olemnab  10678  ennnfonelemrn  12875  mhmima  13408  ghmnsgima  13689  lmtopcnp  14807