Theorem fniunfv 2860

Description: The indexed union of a function's values is the union of its range.

Assertion

Ref	Expression
fniunfv	|- (F Fn A -> U.x e. A (F` x) = U.ran F)

Distinct variable group(s):   x,A   x,F

Proof of Theorem fniunfv

Step	Hyp	Ref	Expression
1		fndm 2723	. . . . . . . . . . 11 |- (F Fn A -> dom F = A)
2	1	eleq2d 1156	. . . . . . . . . 10 |- (F Fn A -> (x e. dom F <-> x e. A))
3		visset 1350	. . . . . . . . . . 11 |- x e. V
4	3	opeldm 2534	. . . . . . . . . 10 |- (<.x, y>. e. F -> x e. dom F)
5	2, 4	syl5bi 183	. . . . . . . . 9 |- (F Fn A -> (<.x, y>. e. F -> x e. A))
6		pm4.71r 482	. . . . . . . . 9 |- ((<.x, y>. e. F -> x e. A) <-> (<.x, y>. e. F <-> (x e. A /\ <.x, y>. e. F)))
7	5, 6	sylib 173	. . . . . . . 8 |- (F Fn A -> (<.x, y>. e. F <-> (x e. A /\ <.x, y>. e. F)))
8		visset 1350	. . . . . . . . . . . 12 |- y e. V
9	8	fnfvop 2856	. . . . . . . . . . 11 |- ((F Fn A /\ x e. A) -> ((F` x) = y <-> <.x, y>. e. F))
10		cleqcom 1103	. . . . . . . . . . 11 |- ((F` x) = y <-> y = (F` x))
11	9, 10	syl5rbbr 413	. . . . . . . . . 10 |- ((F Fn A /\ x e. A) -> (<.x, y>. e. F <-> y = (F` x)))
12	11	exp 291	. . . . . . . . 9 |- (F Fn A -> (x e. A -> (<.x, y>. e. F <-> y = (F` x))))
13	12	pm5.32d 491	. . . . . . . 8 |- (F Fn A -> ((x e. A /\ <.x, y>. e. F) <-> (x e. A /\ y = (F` x))))
14	7, 13	bitrd 406	. . . . . . 7 |- (F Fn A -> (<.x, y>. e. F <-> (x e. A /\ y = (F` x))))
15	14	biexdv 936	. . . . . 6 |- (F Fn A -> (E.x<.x, y>. e. F <-> E.x(x e. A /\ y = (F` x))))
16		df-rex 1206	. . . . . 6 |- (E.x e. A y = (F` x) <-> E.x(x e. A /\ y = (F` x)))
17	15, 16	syl6bbr 416	. . . . 5 |- (F Fn A -> (E.x<.x, y>. e. F <-> E.x e. A y = (F` x)))
18	17	biabdv 1183	. . . 4 |- (F Fn A -> {y | E.x<.x, y>. e. F} = {y | E.x e. A y = (F` x)})
19		dfrn3 2524	. . . 4 |- ran F = {y | E.x<.x, y>. e. F}
20	18, 19	syl5req 1137	. . 3 |- (F Fn A -> {y | E.x e. A y = (F` x)} = ran F)
21	20	unieqd 1929	. 2 |- (F Fn A -> U.{y | E.x e. A y = (F` x)} = U.ran F)
22		fvex 2838	. . 3 |- (F` x) e. V
23	22	dfiun2 2014	. 2 |- U.x e. A (F` x) = U.{y | E.x e. A y = (F` x)}
24	21, 23	syl5eq 1136	1 |- (F Fn A -> U.x e. A (F` x) = U.ran F)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678  {cab 1090   = wceq 1091   e. wcel 1092  E.wrex 1202  <.cop 1810  U.cuni 1919  U.ciun 1994  dom cdm 2410  ran crn 2411   Fn wfn 2417  ` cfv 2422

This theorem is referenced by:  unir1 3511

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-iun 1996  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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