Theorem funiunfv 3866

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 3732	. . . . . 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 3618	. . . . 5 |- {<.y, z>. | (y e. A /\ z = (F` y))} Fn A
4		fniunfv 3865	. . . . 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 3724	. . . . 5 |- (y = x -> (F` y) = (F` x))
8		fvex 3732	. . . . 5 |- (F` x) e. V
9	7, 2, 8	fvopab4 3780	. . . 4 |- (x e. A -> ({<.y, z>. | (y e. A /\ z = (F` y))}` x) = (F` x))
10	9	iuneq2i 2580	. . 3 |- U_x e. A ({<.y, z>. | (y e. A /\ z = (F` y))}` x) = U_x e. A (F` x)
11	6, 10	syl5eqr 1521	. 2 |- (Fun F -> U_x e. A (F` x) = U.ran {<.y, z>. | (y e. A /\ z = (F` y))})
12		visset 1813	. . . . . . . . . . . . . . . . 17 |- z e. V
13	12	funbrfvb 3755	. . . . . . . . . . . . . . . 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 1481	. . . . . . . . . . . . . . . . . . 19 |- ((F` y) = z -> ((F` y) = (/) <-> z = (/)))
16		ndmfv 3745	. . . . . . . . . . . . . . . . . . 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 2285	. . . . . . . . . . . . . . . 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 3750	. . . . . . . . . . . 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 516	. . . . . . . . . 10 |- ((Fun F /\ w e. z) -> ((F` y) = z <-> yFz))
29		eqcom 1477	. . . . . . . . . 10 |- (z = (F` y) <-> (F` y) = z)
30	28, 29	syl5bb 532	. . . . . . . . 9 |- ((Fun F /\ w e. z) -> (z = (F` y) <-> yFz))
31	30	rexbidv 1664	. . . . . . . 8 |- ((Fun F /\ w e. z) -> (E.y e. A z = (F` y) <-> E.y e. A yFz))
32	31	pm5.32da 649	. . . . . . 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 2506	. . . . . . 7 |- (w e. U.(F"A) <-> E.z(w e. z /\ z e. (F"A)))
35	12	elima 3408	. . . . . . . . 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 1051	. . . . . . 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 536	. . . . 5 |- (Fun F -> (E.z(w e. z /\ E.y e. A z = (F` y)) <-> w e. U.(F"A)))
40		eluniab 2513	. . . . 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 532	. . . 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 3354	. . . 4 |- ran {<.y, z>. | (y e. A /\ z = (F` y))} = {z | E.y e. A z = (F` y)}
44	43	unieqi 2511	. . 3 |- U.ran {<.y, z>. | (y e. A /\ z = (F` y))} = U.{z | E.y e. A z = (F` y)}
45	42, 44	syl5eq 1519	. 2 |- (Fun F -> U.ran {<.y, z>. | (y e. A /\ z = (F` y))} = U.(F"A))
46	11, 45	eqtrd 1507	1 |- (Fun F -> U_x e. A (F` x) = U.(F"A))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  E.wrex 1646  (/)c0 2280  U.cuni 2503  U_ciun 2566   class class class wbr 2619  {copab 2666  dom cdm 3170  ran crn 3171  "cima 3173  Fun wfun 3176   Fn wfn 3177  ` cfv 3182

This theorem is referenced by:  eluniima 3867  funiunfvf 3870

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-9 965  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 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

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 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198

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