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Theorem funiunfv 7249
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 6589 . . . 4 (Fun 𝐹 → Fun (𝐹𝐴))
21funfnd 6578 . . 3 (Fun 𝐹 → (𝐹𝐴) Fn dom (𝐹𝐴))
3 fniunfv 7248 . . 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 4475 . . . . 5 (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴))) = (dom (𝐹𝐴) ∪ 𝐴)
6 dmres 6002 . . . . . . 7 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
7 inss1 4227 . . . . . . 7 (𝐴 ∩ dom 𝐹) ⊆ 𝐴
86, 7eqsstri 4015 . . . . . 6 dom (𝐹𝐴) ⊆ 𝐴
9 ssequn1 4179 . . . . . 6 (dom (𝐹𝐴) ⊆ 𝐴 ↔ (dom (𝐹𝐴) ∪ 𝐴) = 𝐴)
108, 9mpbi 229 . . . . 5 (dom (𝐹𝐴) ∪ 𝐴) = 𝐴
115, 10eqtri 2758 . . . 4 (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴))) = 𝐴
12 iuneq1 5012 . . . 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 5096 . . . 4 𝑥 ∈ (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴)))((𝐹𝐴)‘𝑥) = ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥))
15 eldifn 4126 . . . . . . . . 9 (𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴)) → ¬ 𝑥 ∈ dom (𝐹𝐴))
16 ndmfv 6925 . . . . . . . . 9 𝑥 ∈ dom (𝐹𝐴) → ((𝐹𝐴)‘𝑥) = ∅)
1715, 16syl 17 . . . . . . . 8 (𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴)) → ((𝐹𝐴)‘𝑥) = ∅)
1817iuneq2i 5017 . . . . . . 7 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥) = 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))∅
19 iun0 5064 . . . . . . 7 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))∅ = ∅
2018, 19eqtri 2758 . . . . . 6 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥) = ∅
2120uneq2i 4159 . . . . 5 ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥)) = ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ ∅)
22 un0 4389 . . . . 5 ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ ∅) = 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥)
2321, 22eqtri 2758 . . . 4 ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥)) = 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥)
2414, 23eqtri 2758 . . 3 𝑥 ∈ (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴)))((𝐹𝐴)‘𝑥) = 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥)
25 fvres 6909 . . . 4 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
2625iuneq2i 5017 . . 3 𝑥𝐴 ((𝐹𝐴)‘𝑥) = 𝑥𝐴 (𝐹𝑥)
2713, 24, 263eqtr3ri 2767 . 2 𝑥𝐴 (𝐹𝑥) = 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥)
28 df-ima 5688 . . 3 (𝐹𝐴) = ran (𝐹𝐴)
2928unieqi 4920 . 2 (𝐹𝐴) = ran (𝐹𝐴)
304, 27, 293eqtr4g 2795 1 (Fun 𝐹 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1539  wcel 2104  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4321   cuni 4907   ciun 4996  dom cdm 5675  ran crn 5676  cres 5677  cima 5678  Fun wfun 6536   Fn wfn 6537  cfv 6542
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550
This theorem is referenced by:  funiunfvf  7250  eluniima  7251  marypha2lem4  9435  r1limg  9768  r1elssi  9802  r1elss  9803  ackbij2  10240  r1om  10241  ttukeylem6  10511  isacs2  17601  mreacs  17606  acsfn  17607  isacs5  18505  dprdss  19940  dprd2dlem1  19952  dmdprdsplit2lem  19956  uniioombllem3a  25333  uniioombllem4  25335  uniioombllem5  25336  dyadmbl  25349  oldlim  27618  precsexlem10  27901  precsexlem11  27902  mblfinlem1  36828  ovoliunnfl  36833  voliunnfl  36835  uniimafveqt  46347  imasetpreimafvbijlemfv  46368
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