Theorem funiunfv 7285

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 6620	. . . 4 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
2	1	funfnd 6609	. . 3 ⊢ (Fun 𝐹 → (𝐹 ↾ 𝐴) Fn dom (𝐹 ↾ 𝐴))
3		fniunfv 7284	. . 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 4500	. . . . 5 ⊢ (dom (𝐹 ↾ 𝐴) ∪ (𝐴 ∖ dom (𝐹 ↾ 𝐴))) = (dom (𝐹 ↾ 𝐴) ∪ 𝐴)
6		dmres 6041	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
7		inss1 4258	. . . . . . 7 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
8	6, 7	eqsstri 4043	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴
9		ssequn1 4209	. . . . . 6 ⊢ (dom (𝐹 ↾ 𝐴) ⊆ 𝐴 ↔ (dom (𝐹 ↾ 𝐴) ∪ 𝐴) = 𝐴)
10	8, 9	mpbi 230	. . . . 5 ⊢ (dom (𝐹 ↾ 𝐴) ∪ 𝐴) = 𝐴
11	5, 10	eqtri 2768	. . . 4 ⊢ (dom (𝐹 ↾ 𝐴) ∪ (𝐴 ∖ dom (𝐹 ↾ 𝐴))) = 𝐴
12		iuneq1 5031	. . . 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 5117	. . . 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 4155	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴)) → ¬ 𝑥 ∈ dom (𝐹 ↾ 𝐴))
16		ndmfv 6955	. . . . . . . . 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 5036	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))((𝐹 ↾ 𝐴)‘𝑥) = ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))∅
19		iun0 5085	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))∅ = ∅
20	18, 19	eqtri 2768	. . . . . 6 ⊢ ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))((𝐹 ↾ 𝐴)‘𝑥) = ∅
21	20	uneq2i 4188	. . . . 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 4417	. . . . 5 ⊢ (∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥) ∪ ∅) = ∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥)
23	21, 22	eqtri 2768	. . . 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 2768	. . 3 ⊢ ∪ 𝑥 ∈ (dom (𝐹 ↾ 𝐴) ∪ (𝐴 ∖ dom (𝐹 ↾ 𝐴)))((𝐹 ↾ 𝐴)‘𝑥) = ∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥)
25		fvres 6939	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
26	25	iuneq2i 5036	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐴)‘𝑥) = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥)
27	13, 24, 26	3eqtr3ri 2777	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥)
28		df-ima 5713	. . 3 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
29	28	unieqi 4943	. 2 ⊢ ∪ (𝐹 “ 𝐴) = ∪ ran (𝐹 ↾ 𝐴)
30	4, 27, 29	3eqtr4g 2805	1 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   ∈ wcel 2108   ∖ cdif 3973   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ∪ cuni 4931  ∪ ciun 5015  dom cdm 5700  ran crn 5701   ↾ cres 5702   “ cima 5703  Fun wfun 6567   Fn wfn 6568  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581

This theorem is referenced by:  funiunfvf  7286  eluniima  7287  marypha2lem4  9507  r1limg  9840  r1elssi  9874  r1elss  9875  ackbij2  10311  r1om  10312  ttukeylem6  10583  isacs2  17711  mreacs  17716  acsfn  17717  isacs5  18618  dprdss  20073  dprd2dlem1  20085  dmdprdsplit2lem  20089  uniioombllem3a  25638  uniioombllem4  25640  uniioombllem5  25641  dyadmbl  25654  oldlim  27943  precsexlem10  28258  precsexlem11  28259  mblfinlem1  37617  ovoliunnfl  37622  voliunnfl  37624  uniimafveqt  47255  imasetpreimafvbijlemfv  47276