Theorem funiunfv 6506

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. (Contributed by NM, 26-Mar-2006.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

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		funres 5929	. . . 4 |- ( Fun F -> Fun ( F |` A ) )
2		funfn 5918	. . . 4 |- ( Fun ( F |` A ) <-> ( F |` A ) Fn dom ( F |` A ) )
3	1, 2	sylib 208	. . 3 |- ( Fun F -> ( F |` A ) Fn dom ( F |` A ) )
4		fniunfv 6505	. . 3 |- ( ( F |` A ) Fn dom ( F |` A ) -> U_ x e. dom ( F |` A ) ( ( F |` A ) ` x ) = U. ran ( F |` A ) )
5	3, 4	syl 17	. 2 |- ( Fun F -> U_ x e. dom ( F |` A ) ( ( F |` A ) ` x ) = U. ran ( F |` A ) )
6		undif2 4044	. . . . 5 |- ( dom ( F |` A ) u. ( A \ dom ( F |` A ) ) ) = ( dom ( F |` A ) u. A )
7		dmres 5419	. . . . . . 7 |- dom ( F |` A ) = ( A i^i dom F )
8		inss1 3833	. . . . . . 7 |- ( A i^i dom F ) C_ A
9	7, 8	eqsstri 3635	. . . . . 6 |- dom ( F |` A ) C_ A
10		ssequn1 3783	. . . . . 6 |- ( dom ( F |` A ) C_ A <-> ( dom ( F |` A ) u. A ) = A )
11	9, 10	mpbi 220	. . . . 5 |- ( dom ( F |` A ) u. A ) = A
12	6, 11	eqtri 2644	. . . 4 |- ( dom ( F |` A ) u. ( A \ dom ( F |` A ) ) ) = A
13		iuneq1 4534	. . . 4 |- ( ( dom ( F |` A ) u. ( A \ dom ( F |` A ) ) ) = A -> U_ x e. ( dom ( F |` A ) u. ( A \ dom ( F |` A ) ) ) ( ( F |` A ) ` x ) = U_ x e. A ( ( F |` A ) ` x ) )
14	12, 13	ax-mp 5	. . 3 |- U_ x e. ( dom ( F |` A ) u. ( A \ dom ( F |` A ) ) ) ( ( F |` A ) ` x ) = U_ x e. A ( ( F |` A ) ` x )
15		iunxun 4605	. . . 4 |- U_ x e. ( dom ( F |` A ) u. ( A \ dom ( F |` A ) ) ) ( ( F |` A ) ` x ) = ( U_ x e. dom ( F |` A ) ( ( F |` A ) ` x ) u. U_ x e. ( A \ dom ( F |` A ) ) ( ( F |` A ) ` x ) )
16		eldifn 3733	. . . . . . . . 9 |- ( x e. ( A \ dom ( F |` A ) ) -> -. x e. dom ( F |` A ) )
17		ndmfv 6218	. . . . . . . . 9 |- ( -. x e. dom ( F |` A ) -> ( ( F |` A ) ` x ) = (/) )
18	16, 17	syl 17	. . . . . . . 8 |- ( x e. ( A \ dom ( F |` A ) ) -> ( ( F |` A ) ` x ) = (/) )
19	18	iuneq2i 4539	. . . . . . 7 |- U_ x e. ( A \ dom ( F |` A ) ) ( ( F |` A ) ` x ) = U_ x e. ( A \ dom ( F |` A ) ) (/)
20		iun0 4576	. . . . . . 7 |- U_ x e. ( A \ dom ( F |` A ) ) (/) = (/)
21	19, 20	eqtri 2644	. . . . . 6 |- U_ x e. ( A \ dom ( F |` A ) ) ( ( F |` A ) ` x ) = (/)
22	21	uneq2i 3764	. . . . 5 |- ( U_ x e. dom ( F |` A ) ( ( F |` A ) ` x ) u. U_ x e. ( A \ dom ( F |` A ) ) ( ( F |` A ) ` x ) ) = ( U_ x e. dom ( F |` A ) ( ( F |` A ) ` x ) u. (/) )
23		un0 3967	. . . . 5 |- ( U_ x e. dom ( F |` A ) ( ( F |` A ) ` x ) u. (/) ) = U_ x e. dom ( F |` A ) ( ( F |` A ) ` x )
24	22, 23	eqtri 2644	. . . 4 |- ( U_ x e. dom ( F |` A ) ( ( F |` A ) ` x ) u. U_ x e. ( A \ dom ( F |` A ) ) ( ( F |` A ) ` x ) ) = U_ x e. dom ( F |` A ) ( ( F |` A ) ` x )
25	15, 24	eqtri 2644	. . 3 |- U_ x e. ( dom ( F |` A ) u. ( A \ dom ( F |` A ) ) ) ( ( F |` A ) ` x ) = U_ x e. dom ( F |` A ) ( ( F |` A ) ` x )
26		fvres 6207	. . . 4 |- ( x e. A -> ( ( F |` A ) ` x ) = ( F ` x ) )
27	26	iuneq2i 4539	. . 3 |- U_ x e. A ( ( F |` A ) ` x ) = U_ x e. A ( F ` x )
28	14, 25, 27	3eqtr3ri 2653	. 2 |- U_ x e. A ( F ` x ) = U_ x e. dom ( F |` A ) ( ( F |` A ) ` x )
29		df-ima 5127	. . 3 |- ( F " A ) = ran ( F |` A )
30	29	unieqi 4445	. 2 |- U. ( F " A ) = U. ran ( F |` A )
31	5, 28, 30	3eqtr4g 2681	1 |- ( Fun F -> U_ x e. A ( F ` x ) = U. ( F " A ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   U.cuni 4436   U_ciun 4520   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   Fn wfn 5883   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  funiunfvf  6507  eluniima  6508  marypha2lem4  8344  r1limg  8634  r1elssi  8668  r1elss  8669  ackbij2  9065  r1om  9066  ttukeylem6  9336  isacs2  16314  mreacs  16319  acsfn  16320  isacs5  17172  dprdss  18428  dprd2dlem1  18440  dmdprdsplit2lem  18444  uniioombllem3a  23352  uniioombllem4  23354  uniioombllem5  23355  dyadmbl  23368  mblfinlem1  33446  ovoliunnfl  33451  voliunnfl  33453