Theorem funfv 3768

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 3765	. . . . . 6 |- ((F Fn dom F /\ A e. dom F) -> {(F` A)} = (F"{A}))
2		df-fn 3191	. . . . . . 7 |- (F Fn dom F <-> (Fun F /\ dom F = dom F))
3		eqid 1475	. . . . . . 7 |- dom F = dom F
4	2, 3	mpbiran2 729	. . . . . 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 2510	. . . 4 |- ((Fun F /\ A e. dom F) -> U.{(F` A)} = U.(F"{A}))
7		fvex 3730	. . . . 5 |- (F` A) e. V
8	7	unisn 2515	. . . 4 |- U.{(F` A)} = (F` A)
9	6, 8	syl5eqr 1520	. . 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 3743	. . 3 |- (-. A e. dom F -> (F` A) = (/))
12		ndmima 3432	. . . . 5 |- (-. A e. dom F -> (F"{A}) = (/))
13	12	unieqd 2510	. . . 4 |- (-. A e. dom F -> U.(F"{A}) = U.(/))
14		uni0 2523	. . . 4 |- U.(/) = (/)
15	13, 14	syl6eq 1522	. . 3 |- (-. A e. dom F -> U.(F"{A}) = (/))
16	11, 15	eqtr4d 1509	. 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 956   e. wcel 958  (/)c0 2278  {csn 2407  U.cuni 2501  dom cdm 3168  "cima 3171  Fun wfun 3174   Fn wfn 3175  ` cfv 3180

This theorem is referenced by:  funfv2 3769

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2701  ax-pow 2740  ax-pr 2777  ax-un 2864

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2502  df-br 2618  df-opab 2665  df-id 2833  df-xp 3182  df-rel 3183  df-cnv 3184  df-co 3185  df-dm 3186  df-rn 3187  df-res 3188  df-ima 3189  df-fun 3190  df-fn 3191  df-fv 3196

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