Theorem funfvima3 4830

Description: A class including a function contains the function's value in the image of the singleton of the argument.

Assertion

Ref	Expression
funfvima3	|- ((Fun F /\ F C_ G) -> (A e. dom F -> (F` A) e. (G"{A})))

Proof of Theorem funfvima3

Step	Hyp	Ref	Expression
1		ssel 2615	. . . . . 6 |- (F C_ G -> (<.A, (F` A)>. e. F -> <.A, (F` A)>. e. G))
2		funfvop 4776	. . . . . 6 |- ((Fun F /\ A e. dom F) -> <.A, (F` A)>. e. F)
3	1, 2	syl5 20	. . . . 5 |- (F C_ G -> ((Fun F /\ A e. dom F) -> <.A, (F` A)>. e. G))
4	3	imp 377	. . . 4 |- ((F C_ G /\ (Fun F /\ A e. dom F)) -> <.A, (F` A)>. e. G)
5		sneq 3054	. . . . . . . 8 |- (x = A -> {x} = {A})
6	5	imaeq2d 4264	. . . . . . 7 |- (x = A -> (G"{x}) = (G"{A}))
7	6	eleq2d 1964	. . . . . 6 |- (x = A -> ((F` A) e. (G"{x}) <-> (F` A) e. (G"{A})))
8		opeq1 3158	. . . . . . 7 |- (x = A -> <.x, (F` A)>. = <.A, (F` A)>.)
9	8	eleq1d 1963	. . . . . 6 |- (x = A -> (<.x, (F` A)>. e. G <-> <.A, (F` A)>. e. G))
10		visset 2295	. . . . . . 7 |- x e. _V
11		fvex 4689	. . . . . . 7 |- (F` A) e. _V
12	10, 11	elimasn 4289	. . . . . 6 |- ((F` A) e. (G"{x}) <-> <.x, (F` A)>. e. G)
13	7, 9, 12	vtoclbg 2347	. . . . 5 |- (A e. dom F -> ((F` A) e. (G"{A}) <-> <.A, (F` A)>. e. G))
14	13	ad2antll 443	. . . 4 |- ((F C_ G /\ (Fun F /\ A e. dom F)) -> ((F` A) e. (G"{A}) <-> <.A, (F` A)>. e. G))
15	4, 14	mpbird 213	. . 3 |- ((F C_ G /\ (Fun F /\ A e. dom F)) -> (F` A) e. (G"{A}))
16	15	exp32 408	. 2 |- (F C_ G -> (Fun F -> (A e. dom F -> (F` A) e. (G"{A}))))
17	16	impcom 378	1 |- ((Fun F /\ F C_ G) -> (A e. dom F -> (F` A) e. (G"{A})))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   C_ wss 2593  {csn 3044  <.cop 3046  dom cdm 3986  "cima 3989  Fun wfun 3992  ` cfv 3998

This theorem is referenced by:  aceq3 5895

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014

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