Theorem funiunfv 7248

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 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem funiunfv

Step	Hyp	Ref	Expression
1		funres 6579	. . . 4 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
2	1	funfnd 6568	. . 3 ⊢ (Fun 𝐹 → (𝐹 ↾ 𝐴) Fn dom (𝐹 ↾ 𝐴))
3		fniunfv 7247	. . 3 ⊢ ((𝐹 ↾ 𝐴) Fn dom (𝐹 ↾ 𝐴) → ∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥) = ∪ ran (𝐹 ↾ 𝐴))
4	2, 3	syl 18	. 2 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥) = ∪ ran (𝐹 ↾ 𝐴))
5		undif2 4434	. . . . 5 ⊢ (dom (𝐹 ↾ 𝐴) ∪ (𝐴 ∖ dom (𝐹 ↾ 𝐴))) = (dom (𝐹 ↾ 𝐴) ∪ 𝐴)
6		dmres 6009	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
7		inss1 4185	. . . . . . 7 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
8	6, 7	eqsstri 3980	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴
9		ssequn1 4135	. . . . . 6 ⊢ (dom (𝐹 ↾ 𝐴) ⊆ 𝐴 ↔ (dom (𝐹 ↾ 𝐴) ∪ 𝐴) = 𝐴)
10	8, 9	mpbi 233	. . . . 5 ⊢ (dom (𝐹 ↾ 𝐴) ∪ 𝐴) = 𝐴
11	5, 10	eqtri 2785	. . . 4 ⊢ (dom (𝐹 ↾ 𝐴) ∪ (𝐴 ∖ dom (𝐹 ↾ 𝐴))) = 𝐴
12		iuneq1 4971	. . . 4 ⊢ ((dom (𝐹 ↾ 𝐴) ∪ (𝐴 ∖ dom (𝐹 ↾ 𝐴))) = 𝐴 → ∪ 𝑥 ∈ (dom (𝐹 ↾ 𝐴) ∪ (𝐴 ∖ dom (𝐹 ↾ 𝐴)))((𝐹 ↾ 𝐴)‘𝑥) = ∪ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐴)‘𝑥))
13	11, 12	ax-mp 5	. . 3 ⊢ ∪ 𝑥 ∈ (dom (𝐹 ↾ 𝐴) ∪ (𝐴 ∖ dom (𝐹 ↾ 𝐴)))((𝐹 ↾ 𝐴)‘𝑥) = ∪ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐴)‘𝑥)
14		iunxun 5058	. . . 4 ⊢ ∪ 𝑥 ∈ (dom (𝐹 ↾ 𝐴) ∪ (𝐴 ∖ dom (𝐹 ↾ 𝐴)))((𝐹 ↾ 𝐴)‘𝑥) = (∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥) ∪ ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))((𝐹 ↾ 𝐴)‘𝑥))
15		eldifn 4082	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴)) → ¬ 𝑥 ∈ dom (𝐹 ↾ 𝐴))
16		ndmfv 6914	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ dom (𝐹 ↾ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = ∅)
17	15, 16	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝑥) = ∅)
18	17	iuneq2i 4976	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))((𝐹 ↾ 𝐴)‘𝑥) = ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))∅
19		iun0 5024	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))∅ = ∅
20	18, 19	eqtri 2785	. . . . . 6 ⊢ ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))((𝐹 ↾ 𝐴)‘𝑥) = ∅
21	20	uneq2i 4115	. . . . 5 ⊢ (∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥) ∪ ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))((𝐹 ↾ 𝐴)‘𝑥)) = (∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥) ∪ ∅)
22		un0 4347	. . . . 5 ⊢ (∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥) ∪ ∅) = ∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥)
23	21, 22	eqtri 2785	. . . 4 ⊢ (∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥) ∪ ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))((𝐹 ↾ 𝐴)‘𝑥)) = ∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥)
24	14, 23	eqtri 2785	. . 3 ⊢ ∪ 𝑥 ∈ (dom (𝐹 ↾ 𝐴) ∪ (𝐴 ∖ dom (𝐹 ↾ 𝐴)))((𝐹 ↾ 𝐴)‘𝑥) = ∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥)
25		fvres 6901	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
26	25	iuneq2i 4976	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐴)‘𝑥) = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥)
27	13, 24, 26	3eqtr3ri 2794	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥)
28		df-ima 5672	. . 3 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
29	28	unieqi 4882	. 2 ⊢ ∪ (𝐹 “ 𝐴) = ∪ ran (𝐹 ↾ 𝐴)
30	4, 27, 29	3eqtr4g 2822	1 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2145   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  ∪ cuni 4870  ∪ ciun 4954  dom cdm 5659  ran crn 5660   ↾ cres 5661   “ cima 5662  Fun wfun 6531   Fn wfn 6532  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545

This theorem is used by:  funiunfvf  7249  eluniima  7250  marypha2lem4  9411  r1limg  9756  r1elssi  9790  r1elss  9791  ackbij2  10247  r1om  10248  ttukeylem6  10519  isacs2  17745  mreacs  17750  acsfn  17751  isacs5  18640  dprdss  20159  dprd2dlem1  20171  dmdprdsplit2lem  20175  uniioombllem3a  25813  uniioombllem4  25815  uniioombllem5  25816  dyadmbl  25829  oldlim  28150  precsexlem10  28479  precsexlem11  28480  r1omfv  35605  ttcmin  37102  dfttc2g  37112  mblfinlem1  38393  ovoliunnfl  38398  voliunnfl  38400  uniimafveqt  48268  imasetpreimafvbijlemfv  48289