Theorem funiunfv 7184

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 6524	. . . 4 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
2	1	funfnd 6513	. . 3 ⊢ (Fun 𝐹 → (𝐹 ↾ 𝐴) Fn dom (𝐹 ↾ 𝐴))
3		fniunfv 7183	. . 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 4428	. . . . 5 ⊢ (dom (𝐹 ↾ 𝐴) ∪ (𝐴 ∖ dom (𝐹 ↾ 𝐴))) = (dom (𝐹 ↾ 𝐴) ∪ 𝐴)
6		dmres 5963	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
7		inss1 4188	. . . . . . 7 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
8	6, 7	eqsstri 3982	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴
9		ssequn1 4137	. . . . . 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 2752	. . . 4 ⊢ (dom (𝐹 ↾ 𝐴) ∪ (𝐴 ∖ dom (𝐹 ↾ 𝐴))) = 𝐴
12		iuneq1 4958	. . . 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 5043	. . . 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 4083	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴)) → ¬ 𝑥 ∈ dom (𝐹 ↾ 𝐴))
16		ndmfv 6855	. . . . . . . . 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 4963	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))((𝐹 ↾ 𝐴)‘𝑥) = ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))∅
19		iun0 5011	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))∅ = ∅
20	18, 19	eqtri 2752	. . . . . 6 ⊢ ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))((𝐹 ↾ 𝐴)‘𝑥) = ∅
21	20	uneq2i 4116	. . . . 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 4345	. . . . 5 ⊢ (∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥) ∪ ∅) = ∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥)
23	21, 22	eqtri 2752	. . . 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 2752	. . 3 ⊢ ∪ 𝑥 ∈ (dom (𝐹 ↾ 𝐴) ∪ (𝐴 ∖ dom (𝐹 ↾ 𝐴)))((𝐹 ↾ 𝐴)‘𝑥) = ∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥)
25		fvres 6841	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
26	25	iuneq2i 4963	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐴)‘𝑥) = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥)
27	13, 24, 26	3eqtr3ri 2761	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ 𝑥 ∈ dom (𝐹 ↾ 𝐴)((𝐹 ↾ 𝐴)‘𝑥)
28		df-ima 5632	. . 3 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
29	28	unieqi 4870	. 2 ⊢ ∪ (𝐹 “ 𝐴) = ∪ ran (𝐹 ↾ 𝐴)
30	4, 27, 29	3eqtr4g 2789	1 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109   ∖ cdif 3900   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  ∪ cuni 4858  ∪ ciun 4941  dom cdm 5619  ran crn 5620   ↾ cres 5621   “ cima 5622  Fun wfun 6476   Fn wfn 6477  ‘cfv 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-fv 6490

This theorem is referenced by:  funiunfvf  7185  eluniima  7186  marypha2lem4  9328  r1limg  9667  r1elssi  9701  r1elss  9702  ackbij2  10136  r1om  10137  ttukeylem6  10408  isacs2  17559  mreacs  17564  acsfn  17565  isacs5  18454  dprdss  19910  dprd2dlem1  19922  dmdprdsplit2lem  19926  uniioombllem3a  25483  uniioombllem4  25485  uniioombllem5  25486  dyadmbl  25499  oldlim  27803  precsexlem10  28125  precsexlem11  28126  mblfinlem1  37657  ovoliunnfl  37662  voliunnfl  37664  uniimafveqt  47385  imasetpreimafvbijlemfv  47406