Theorem funiunfv 7061

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 6422	. . . 4 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
2	1	funfnd 6411	. . 3 ⊢ (Fun 𝐹 → (𝐹 ↾ 𝐴) Fn dom (𝐹 ↾ 𝐴))
3		fniunfv 7060	. . 3 ⊢ ((𝐹 ↾ 𝐴) Fn dom (𝐹 ↾ 𝐴) → ∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥) = ∪ ran (𝐹 ↾ 𝐴))
4	2, 3	syl 17	. 2 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥) = ∪ ran (𝐹 ↾ 𝐴))
5		undif2 4391	. . . . 5 ⊢ (dom (𝐹 ↾ 𝐴) ∪ (𝐴 ∖ dom (𝐹 ↾ 𝐴))) = (dom (𝐹 ↾ 𝐴) ∪ 𝐴)
6		dmres 5873	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
7		inss1 4143	. . . . . . 7 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
8	6, 7	eqsstri 3935	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴
9		ssequn1 4094	. . . . . 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 2765	. . . 4 ⊢ (dom (𝐹 ↾ 𝐴) ∪ (𝐴 ∖ dom (𝐹 ↾ 𝐴))) = 𝐴
12		iuneq1 4920	. . . 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 5002	. . . 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 4042	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴)) → ¬ 𝑥 ∈ dom (𝐹 ↾ 𝐴))
16		ndmfv 6747	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ dom (𝐹 ↾ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = ∅)
17	15, 16	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝑥) = ∅)
18	17	iuneq2i 4925	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))((𝐹 ↾ 𝐴)‘𝑥) = ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))∅
19		iun0 4970	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))∅ = ∅
20	18, 19	eqtri 2765	. . . . . 6 ⊢ ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))((𝐹 ↾ 𝐴)‘𝑥) = ∅
21	20	uneq2i 4074	. . . . 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 4305	. . . . 5 ⊢ (∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥) ∪ ∅) = ∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥)
23	21, 22	eqtri 2765	. . . 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 2765	. . 3 ⊢ ∪ 𝑥 ∈ (dom (𝐹 ↾ 𝐴) ∪ (𝐴 ∖ dom (𝐹 ↾ 𝐴)))((𝐹 ↾ 𝐴)‘𝑥) = ∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥)
25		fvres 6736	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
26	25	iuneq2i 4925	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐴)‘𝑥) = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥)
27	13, 24, 26	3eqtr3ri 2774	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥)
28		df-ima 5564	. . 3 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
29	28	unieqi 4832	. 2 ⊢ ∪ (𝐹 “ 𝐴) = ∪ ran (𝐹 ↾ 𝐴)
30	4, 27, 29	3eqtr4g 2803	1 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1543   ∈ wcel 2110   ∖ cdif 3863   ∪ cun 3864   ∩ cin 3865   ⊆ wss 3866  ∅c0 4237  ∪ cuni 4819  ∪ ciun 4904  dom cdm 5551  ran crn 5552   ↾ cres 5553   “ cima 5554  Fun wfun 6374   Fn wfn 6375  ‘cfv 6380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-fv 6388

This theorem is referenced by:  funiunfvf  7062  eluniima  7063  marypha2lem4  9054  r1limg  9387  r1elssi  9421  r1elss  9422  ackbij2  9857  r1om  9858  ttukeylem6  10128  isacs2  17156  mreacs  17161  acsfn  17162  isacs5  18054  dprdss  19416  dprd2dlem1  19428  dmdprdsplit2lem  19432  uniioombllem3a  24481  uniioombllem4  24483  uniioombllem5  24484  dyadmbl  24497  oldlim  33806  mblfinlem1  35551  ovoliunnfl  35556  voliunnfl  35558  uniimafveqt  44506  imasetpreimafvbijlemfv  44527