MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funiunfv Structured version   Visualization version   GIF version

Theorem funiunfv 6657
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 6078 . . . 4 (Fun 𝐹 → Fun (𝐹𝐴))
2 funfn 6067 . . . 4 (Fun (𝐹𝐴) ↔ (𝐹𝐴) Fn dom (𝐹𝐴))
31, 2sylib 208 . . 3 (Fun 𝐹 → (𝐹𝐴) Fn dom (𝐹𝐴))
4 fniunfv 6656 . . 3 ((𝐹𝐴) Fn dom (𝐹𝐴) → 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) = ran (𝐹𝐴))
53, 4syl 17 . 2 (Fun 𝐹 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) = ran (𝐹𝐴))
6 undif2 4176 . . . . 5 (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴))) = (dom (𝐹𝐴) ∪ 𝐴)
7 dmres 5565 . . . . . . 7 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
8 inss1 3964 . . . . . . 7 (𝐴 ∩ dom 𝐹) ⊆ 𝐴
97, 8eqsstri 3764 . . . . . 6 dom (𝐹𝐴) ⊆ 𝐴
10 ssequn1 3914 . . . . . 6 (dom (𝐹𝐴) ⊆ 𝐴 ↔ (dom (𝐹𝐴) ∪ 𝐴) = 𝐴)
119, 10mpbi 220 . . . . 5 (dom (𝐹𝐴) ∪ 𝐴) = 𝐴
126, 11eqtri 2770 . . . 4 (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴))) = 𝐴
13 iuneq1 4674 . . . 4 ((dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴))) = 𝐴 𝑥 ∈ (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴)))((𝐹𝐴)‘𝑥) = 𝑥𝐴 ((𝐹𝐴)‘𝑥))
1412, 13ax-mp 5 . . 3 𝑥 ∈ (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴)))((𝐹𝐴)‘𝑥) = 𝑥𝐴 ((𝐹𝐴)‘𝑥)
15 iunxun 4745 . . . 4 𝑥 ∈ (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴)))((𝐹𝐴)‘𝑥) = ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥))
16 eldifn 3864 . . . . . . . . 9 (𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴)) → ¬ 𝑥 ∈ dom (𝐹𝐴))
17 ndmfv 6367 . . . . . . . . 9 𝑥 ∈ dom (𝐹𝐴) → ((𝐹𝐴)‘𝑥) = ∅)
1816, 17syl 17 . . . . . . . 8 (𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴)) → ((𝐹𝐴)‘𝑥) = ∅)
1918iuneq2i 4679 . . . . . . 7 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥) = 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))∅
20 iun0 4716 . . . . . . 7 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))∅ = ∅
2119, 20eqtri 2770 . . . . . 6 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥) = ∅
2221uneq2i 3895 . . . . 5 ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥)) = ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ ∅)
23 un0 4098 . . . . 5 ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ ∅) = 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥)
2422, 23eqtri 2770 . . . 4 ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥)) = 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥)
2515, 24eqtri 2770 . . 3 𝑥 ∈ (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴)))((𝐹𝐴)‘𝑥) = 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥)
26 fvres 6356 . . . 4 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
2726iuneq2i 4679 . . 3 𝑥𝐴 ((𝐹𝐴)‘𝑥) = 𝑥𝐴 (𝐹𝑥)
2814, 25, 273eqtr3ri 2779 . 2 𝑥𝐴 (𝐹𝑥) = 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥)
29 df-ima 5267 . . 3 (𝐹𝐴) = ran (𝐹𝐴)
3029unieqi 4585 . 2 (𝐹𝐴) = ran (𝐹𝐴)
315, 28, 303eqtr4g 2807 1 (Fun 𝐹 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1620  wcel 2127  cdif 3700  cun 3701  cin 3702  wss 3703  c0 4046   cuni 4576   ciun 4660  dom cdm 5254  ran crn 5255  cres 5256  cima 5257  Fun wfun 6031   Fn wfn 6032  cfv 6037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-fv 6045
This theorem is referenced by:  funiunfvf  6658  eluniima  6659  marypha2lem4  8497  r1limg  8795  r1elssi  8829  r1elss  8830  ackbij2  9228  r1om  9229  ttukeylem6  9499  isacs2  16486  mreacs  16491  acsfn  16492  isacs5  17344  dprdss  18599  dprd2dlem1  18611  dmdprdsplit2lem  18615  uniioombllem3a  23523  uniioombllem4  23525  uniioombllem5  23526  dyadmbl  23539  mblfinlem1  33728  ovoliunnfl  33733  voliunnfl  33735
  Copyright terms: Public domain W3C validator