Theorem funfv 4731

Description: A simplified expression for the value of a function when we know it's a function.

Assertion

Ref	Expression
funfv	|- (Fun F -> (F` A) = U.(F"{A}))

Proof of Theorem funfv

Step	Hyp	Ref	Expression
1		fnsnfv 4728	. . . . . 6 |- ((F Fn dom F /\ A e. dom F) -> {(F` A)} = (F"{A}))
2		df-fn 4009	. . . . . . 7 |- (F Fn dom F <-> (Fun F /\ dom F = dom F))
3		eqid 1884	. . . . . . 7 |- dom F = dom F
4	2, 3	mpbiran2 799	. . . . . 6 |- (F Fn dom F <-> Fun F)
5	1, 4	sylanbr 499	. . . . 5 |- ((Fun F /\ A e. dom F) -> {(F` A)} = (F"{A}))
6	5	unieqd 3188	. . . 4 |- ((Fun F /\ A e. dom F) -> U.{(F` A)} = U.(F"{A}))
7		fvex 4689	. . . . 5 |- (F` A) e. _V
8	7	unisn 3193	. . . 4 |- U.{(F` A)} = (F` A)
9	6, 8	syl5eqr 1942	. . 3 |- ((Fun F /\ A e. dom F) -> (F` A) = U.(F"{A}))
10	9	ex 402	. 2 |- (Fun F -> (A e. dom F -> (F` A) = U.(F"{A})))
11		ndmfv 4702	. . 3 |- (-. A e. dom F -> (F` A) = (/))
12		ndmima 4300	. . . . 5 |- (-. A e. dom F -> (F"{A}) = (/))
13	12	unieqd 3188	. . . 4 |- (-. A e. dom F -> U.(F"{A}) = U.(/))
14		uni0 3205	. . . 4 |- U.(/) = (/)
15	13, 14	syl6eq 1944	. . 3 |- (-. A e. dom F -> U.(F"{A}) = (/))
16	11, 15	eqtr4d 1928	. 2 |- (-. A e. dom F -> (F` A) = U.(F"{A}))
17	10, 16	pm2.61d1 142	1 |- (Fun F -> (F` A) = U.(F"{A}))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  (/)c0 2875  {csn 3044  U.cuni 3177  dom cdm 3986  "cima 3989  Fun wfun 3992   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  funfv2 4732  dffv2 4734  valfunun 14460

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-ral 2109  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-rel 4001  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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