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Theorem funiunfv 7195
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 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem funiunfv
StepHypRef Expression
1 funres 6543 . . . 4 (Fun 𝐹 → Fun (𝐹𝐴))
21funfnd 6532 . . 3 (Fun 𝐹 → (𝐹𝐴) Fn dom (𝐹𝐴))
3 fniunfv 7194 . . 3 ((𝐹𝐴) Fn dom (𝐹𝐴) → 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) = ran (𝐹𝐴))
42, 3syl 17 . 2 (Fun 𝐹 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) = ran (𝐹𝐴))
5 undif2 4436 . . . . 5 (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴))) = (dom (𝐹𝐴) ∪ 𝐴)
6 dmres 5959 . . . . . . 7 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
7 inss1 4188 . . . . . . 7 (𝐴 ∩ dom 𝐹) ⊆ 𝐴
86, 7eqsstri 3978 . . . . . 6 dom (𝐹𝐴) ⊆ 𝐴
9 ssequn1 4140 . . . . . 6 (dom (𝐹𝐴) ⊆ 𝐴 ↔ (dom (𝐹𝐴) ∪ 𝐴) = 𝐴)
108, 9mpbi 229 . . . . 5 (dom (𝐹𝐴) ∪ 𝐴) = 𝐴
115, 10eqtri 2764 . . . 4 (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴))) = 𝐴
12 iuneq1 4970 . . . 4 ((dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴))) = 𝐴 𝑥 ∈ (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴)))((𝐹𝐴)‘𝑥) = 𝑥𝐴 ((𝐹𝐴)‘𝑥))
1311, 12ax-mp 5 . . 3 𝑥 ∈ (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴)))((𝐹𝐴)‘𝑥) = 𝑥𝐴 ((𝐹𝐴)‘𝑥)
14 iunxun 5054 . . . 4 𝑥 ∈ (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴)))((𝐹𝐴)‘𝑥) = ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥))
15 eldifn 4087 . . . . . . . . 9 (𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴)) → ¬ 𝑥 ∈ dom (𝐹𝐴))
16 ndmfv 6877 . . . . . . . . 9 𝑥 ∈ dom (𝐹𝐴) → ((𝐹𝐴)‘𝑥) = ∅)
1715, 16syl 17 . . . . . . . 8 (𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴)) → ((𝐹𝐴)‘𝑥) = ∅)
1817iuneq2i 4975 . . . . . . 7 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥) = 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))∅
19 iun0 5022 . . . . . . 7 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))∅ = ∅
2018, 19eqtri 2764 . . . . . 6 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥) = ∅
2120uneq2i 4120 . . . . 5 ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥)) = ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ ∅)
22 un0 4350 . . . . 5 ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ ∅) = 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥)
2321, 22eqtri 2764 . . . 4 ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥)) = 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥)
2414, 23eqtri 2764 . . 3 𝑥 ∈ (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴)))((𝐹𝐴)‘𝑥) = 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥)
25 fvres 6861 . . . 4 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
2625iuneq2i 4975 . . 3 𝑥𝐴 ((𝐹𝐴)‘𝑥) = 𝑥𝐴 (𝐹𝑥)
2713, 24, 263eqtr3ri 2773 . 2 𝑥𝐴 (𝐹𝑥) = 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥)
28 df-ima 5646 . . 3 (𝐹𝐴) = ran (𝐹𝐴)
2928unieqi 4878 . 2 (𝐹𝐴) = ran (𝐹𝐴)
304, 27, 293eqtr4g 2801 1 (Fun 𝐹 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1541  wcel 2106  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4282   cuni 4865   ciun 4954  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  Fun wfun 6490   Fn wfn 6491  cfv 6496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-fv 6504
This theorem is referenced by:  funiunfvf  7196  eluniima  7197  marypha2lem4  9374  r1limg  9707  r1elssi  9741  r1elss  9742  ackbij2  10179  r1om  10180  ttukeylem6  10450  isacs2  17533  mreacs  17538  acsfn  17539  isacs5  18437  dprdss  19808  dprd2dlem1  19820  dmdprdsplit2lem  19824  uniioombllem3a  24948  uniioombllem4  24950  uniioombllem5  24951  dyadmbl  24964  oldlim  27216  mblfinlem1  36115  ovoliunnfl  36120  voliunnfl  36122  uniimafveqt  45563  imasetpreimafvbijlemfv  45584
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