Theorem funfv 3755

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 3752	. . . . . 6 |- ((F Fn dom F /\ A e. dom F) -> {(F` A)} = (F"{A}))
2		df-fn 3183	. . . . . . 7 |- (F Fn dom F <-> (Fun F /\ dom F = dom F))
3		eqid 1468	. . . . . . 7 |- dom F = dom F
4	2, 3	mpbiran2 727	. . . . . 6 |- (F Fn dom F <-> Fun F)
5	1, 4	sylanbr 450	. . . . 5 |- ((Fun F /\ A e. dom F) -> {(F` A)} = (F"{A}))
6	5	unieqd 2502	. . . 4 |- ((Fun F /\ A e. dom F) -> U.{(F` A)} = U.(F"{A}))
7		fvex 3717	. . . . 5 |- (F` A) e. V
8	7	unisn 2507	. . . 4 |- U.{(F` A)} = (F` A)
9	6, 8	syl5eqr 1513	. . 3 |- ((Fun F /\ A e. dom F) -> (F` A) = U.(F"{A}))
10	9	ex 373	. 2 |- (Fun F -> (A e. dom F -> (F` A) = U.(F"{A})))
11		ndmfv 3730	. . 3 |- (-. A e. dom F -> (F` A) = (/))
12		ndmima 3418	. . . . 5 |- (-. A e. dom F -> (F"{A}) = (/))
13	12	unieqd 2502	. . . 4 |- (-. A e. dom F -> U.(F"{A}) = U.(/))
14		uni0 2515	. . . 4 |- U.(/) = (/)
15	13, 14	syl6eq 1515	. . 3 |- (-. A e. dom F -> U.(F"{A}) = (/))
16	11, 15	eqtr4d 1502	. 2 |- (-. A e. dom F -> (F` A) = U.(F"{A}))
17	10, 16	pm2.61d1 128	1 |- (Fun F -> (F` A) = U.(F"{A}))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  (/)c0 2270  {csn 2399  U.cuni 2493  dom cdm 3160  "cima 3163  Fun wfun 3166   Fn wfn 3167  ` cfv 3172

This theorem is referenced by:  funfv2 3756

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188

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