Theorem funiunfv 7204

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 6542	. . . 4 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
2	1	funfnd 6531	. . 3 ⊢ (Fun 𝐹 → (𝐹 ↾ 𝐴) Fn dom (𝐹 ↾ 𝐴))
3		fniunfv 7203	. . 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 4436	. . . . 5 ⊢ (dom (𝐹 ↾ 𝐴) ∪ (𝐴 ∖ dom (𝐹 ↾ 𝐴))) = (dom (𝐹 ↾ 𝐴) ∪ 𝐴)
6		dmres 5972	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
7		inss1 4196	. . . . . . 7 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
8	6, 7	eqsstri 3990	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴
9		ssequn1 4145	. . . . . 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 4968	. . . 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 5053	. . . 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 4091	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴)) → ¬ 𝑥 ∈ dom (𝐹 ↾ 𝐴))
16		ndmfv 6875	. . . . . . . . 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 4973	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))((𝐹 ↾ 𝐴)‘𝑥) = ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))∅
19		iun0 5021	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))∅ = ∅
20	18, 19	eqtri 2752	. . . . . 6 ⊢ ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹 ↾ 𝐴))((𝐹 ↾ 𝐴)‘𝑥) = ∅
21	20	uneq2i 4124	. . . . 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 4353	. . . . 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 6859	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
26	25	iuneq2i 4973	. . 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 5644	. . 3 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
29	28	unieqi 4879	. 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 3908   ∪ cun 3909   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  ∪ cuni 4867  ∪ ciun 4951  dom cdm 5631  ran crn 5632   ↾ cres 5633   “ cima 5634  Fun wfun 6493   Fn wfn 6494  ‘cfv 6499

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 5246  ax-nul 5256  ax-pr 5382

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 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-fv 6507

This theorem is referenced by:  funiunfvf  7205  eluniima  7206  marypha2lem4  9365  r1limg  9700  r1elssi  9734  r1elss  9735  ackbij2  10171  r1om  10172  ttukeylem6  10443  isacs2  17594  mreacs  17599  acsfn  17600  isacs5  18489  dprdss  19945  dprd2dlem1  19957  dmdprdsplit2lem  19961  uniioombllem3a  25518  uniioombllem4  25520  uniioombllem5  25521  dyadmbl  25534  oldlim  27836  precsexlem10  28158  precsexlem11  28159  mblfinlem1  37644  ovoliunnfl  37649  voliunnfl  37651  uniimafveqt  47375  imasetpreimafvbijlemfv  47396