Theorem funiunfv 3863

Description: The indexed union of a function's values is the union of its image under the index class.

Note: This theorem depends on the fact that our function value is the empty set outside of its domain. If the antecedent is changed to F Fn A, the theorem can be proved without this dependency.

Assertion

Ref	Expression
funiunfv	|- (Fun F -> U_x e. A (F` x) = U.(F"A))

Distinct variable groups:   x,A   x,F

Proof of Theorem funiunfv

Step	Hyp	Ref	Expression
1		fvex 3729	. . . . . 6 |- (F` y) e. V
2		eqid 1475	. . . . . 6 |- {<.y, z>. | (y e. A /\ z = (F` y))} = {<.y, z>. | (y e. A /\ z = (F` y))}
3	1, 2	fnopab2 3615	. . . . 5 |- {<.y, z>. | (y e. A /\ z = (F` y))} Fn A
4		fniunfv 3862	. . . . 5 |- ({<.y, z>. | (y e. A /\ z = (F` y))} Fn A -> U_x e. A ({<.y, z>. | (y e. A /\ z = (F` y))}` x) = U.ran {<.y, z>. | (y e. A /\ z = (F` y))})
5	3, 4	ax-mp 7	. . . 4 |- U_x e. A ({<.y, z>. | (y e. A /\ z = (F` y))}` x) = U.ran {<.y, z>. | (y e. A /\ z = (F` y))}
6	5	a1i 8	. . 3 |- (Fun F -> U_x e. A ({<.y, z>. | (y e. A /\ z = (F` y))}` x) = U.ran {<.y, z>. | (y e. A /\ z = (F` y))})
7		fveq2 3721	. . . . 5 |- (y = x -> (F` y) = (F` x))
8		fvex 3729	. . . . 5 |- (F` x) e. V
9	7, 2, 8	fvopab4 3777	. . . 4 |- (x e. A -> ({<.y, z>. | (y e. A /\ z = (F` y))}` x) = (F` x))
10	9	iuneq2i 2577	. . 3 |- U_x e. A ({<.y, z>. | (y e. A /\ z = (F` y))}` x) = U_x e. A (F` x)
11	6, 10	syl5eqr 1520	. 2 |- (Fun F -> U_x e. A (F` x) = U.ran {<.y, z>. | (y e. A /\ z = (F` y))})
12		visset 1811	. . . . . . . . . . . . . . . . 17 |- z e. V
13	12	funbrfvb 3752	. . . . . . . . . . . . . . . 16 |- ((Fun F /\ y e. dom F) -> ((F` y) = z <-> yFz))
14	13	biimpd 153	. . . . . . . . . . . . . . 15 |- ((Fun F /\ y e. dom F) -> ((F` y) = z -> yFz))
15		eqeq1 1480	. . . . . . . . . . . . . . . . . . 19 |- ((F` y) = z -> ((F` y) = (/) <-> z = (/)))
16		ndmfv 3742	. . . . . . . . . . . . . . . . . . 19 |- (-. y e. dom F -> (F` y) = (/))
17	15, 16	syl5bi 208	. . . . . . . . . . . . . . . . . 18 |- ((F` y) = z -> (-. y e. dom F -> z = (/)))
18	17	con1d 93	. . . . . . . . . . . . . . . . 17 |- ((F` y) = z -> (-. z = (/) -> y e. dom F))
19	18	impcom 351	. . . . . . . . . . . . . . . 16 |- ((-. z = (/) /\ (F` y) = z) -> y e. dom F)
20		n0i 2283	. . . . . . . . . . . . . . . 16 |- (w e. z -> -. z = (/))
21	19, 20	sylan 448	. . . . . . . . . . . . . . 15 |- ((w e. z /\ (F` y) = z) -> y e. dom F)
22	14, 21	sylan2 451	. . . . . . . . . . . . . 14 |- ((Fun F /\ (w e. z /\ (F` y) = z)) -> ((F` y) = z -> yFz))
23	22	anassrs 441	. . . . . . . . . . . . 13 |- (((Fun F /\ w e. z) /\ (F` y) = z) -> ((F` y) = z -> yFz))
24	23	ex 373	. . . . . . . . . . . 12 |- ((Fun F /\ w e. z) -> ((F` y) = z -> ((F` y) = z -> yFz)))
25	24	pm2.43d 65	. . . . . . . . . . 11 |- ((Fun F /\ w e. z) -> ((F` y) = z -> yFz))
26	12	funbrfv 3747	. . . . . . . . . . . 12 |- (Fun F -> (yFz -> (F` y) = z))
27	26	adantr 389	. . . . . . . . . . 11 |- ((Fun F /\ w e. z) -> (yFz -> (F` y) = z))
28	25, 27	impbid 515	. . . . . . . . . 10 |- ((Fun F /\ w e. z) -> ((F` y) = z <-> yFz))
29		eqcom 1476	. . . . . . . . . 10 |- (z = (F` y) <-> (F` y) = z)
30	28, 29	syl5bb 531	. . . . . . . . 9 |- ((Fun F /\ w e. z) -> (z = (F` y) <-> yFz))
31	30	rexbidv 1663	. . . . . . . 8 |- ((Fun F /\ w e. z) -> (E.y e. A z = (F` y) <-> E.y e. A yFz))
32	31	pm5.32da 648	. . . . . . 7 |- (Fun F -> ((w e. z /\ E.y e. A z = (F` y)) <-> (w e. z /\ E.y e. A yFz)))
33	32	exbidv 1279	. . . . . 6 |- (Fun F -> (E.z(w e. z /\ E.y e. A z = (F` y)) <-> E.z(w e. z /\ E.y e. A yFz)))
34		eluni 2503	. . . . . . 7 |- (w e. U.(F"A) <-> E.z(w e. z /\ z e. (F"A)))
35	12	elima 3405	. . . . . . . . 9 |- (z e. (F"A) <-> E.y e. A yFz)
36	35	anbi2i 480	. . . . . . . 8 |- ((w e. z /\ z e. (F"A)) <-> (w e. z /\ E.y e. A yFz))
37	36	exbii 1050	. . . . . . 7 |- (E.z(w e. z /\ z e. (F"A)) <-> E.z(w e. z /\ E.y e. A yFz))
38	34, 37	bitr2 174	. . . . . 6 |- (E.z(w e. z /\ E.y e. A yFz) <-> w e. U.(F"A))
39	33, 38	syl6bb 535	. . . . 5 |- (Fun F -> (E.z(w e. z /\ E.y e. A z = (F` y)) <-> w e. U.(F"A)))
40		eluniab 2510	. . . . 5 |- (w e. U.{z | E.y e. A z = (F` y)} <-> E.z(w e. z /\ E.y e. A z = (F` y)))
41	39, 40	syl5bb 531	. . . 4 |- (Fun F -> (w e. U.{z | E.y e. A z = (F` y)} <-> w e. U.(F"A)))
42	41	eqrdv 1473	. . 3 |- (Fun F -> U.{z | E.y e. A z = (F` y)} = U.(F"A))
43		rnopab2 3351	. . . 4 |- ran {<.y, z>. | (y e. A /\ z = (F` y))} = {z | E.y e. A z = (F` y)}
44	43	unieqi 2508	. . 3 |- U.ran {<.y, z>. | (y e. A /\ z = (F` y))} = U.{z | E.y e. A z = (F` y)}
45	42, 44	syl5eq 1518	. 2 |- (Fun F -> U.ran {<.y, z>. | (y e. A /\ z = (F` y))} = U.(F"A))
46	11, 45	eqtrd 1506	1 |- (Fun F -> U_x e. A (F` x) = U.(F"A))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  E.wex 979  {cab 1463  E.wrex 1645  (/)c0 2278  U.cuni 2500  U_ciun 2563   class class class wbr 2616  {copab 2663  dom cdm 3167  ran crn 3168  "cima 3170  Fun wfun 3173   Fn wfn 3174  ` cfv 3179

This theorem is referenced by:  eluniima 3864  funiunfvf 3867

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  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 2501  df-iun 2565  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-fv 3195

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