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Theorem funiunfv 6471
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 5897 . . . 4 (Fun 𝐹 → Fun (𝐹𝐴))
2 funfn 5887 . . . 4 (Fun (𝐹𝐴) ↔ (𝐹𝐴) Fn dom (𝐹𝐴))
31, 2sylib 208 . . 3 (Fun 𝐹 → (𝐹𝐴) Fn dom (𝐹𝐴))
4 fniunfv 6470 . . 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 4022 . . . . 5 (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴))) = (dom (𝐹𝐴) ∪ 𝐴)
7 dmres 5388 . . . . . . 7 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
8 inss1 3817 . . . . . . 7 (𝐴 ∩ dom 𝐹) ⊆ 𝐴
97, 8eqsstri 3620 . . . . . 6 dom (𝐹𝐴) ⊆ 𝐴
10 ssequn1 3767 . . . . . 6 (dom (𝐹𝐴) ⊆ 𝐴 ↔ (dom (𝐹𝐴) ∪ 𝐴) = 𝐴)
119, 10mpbi 220 . . . . 5 (dom (𝐹𝐴) ∪ 𝐴) = 𝐴
126, 11eqtri 2643 . . . 4 (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴))) = 𝐴
13 iuneq1 4507 . . . 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 4578 . . . 4 𝑥 ∈ (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴)))((𝐹𝐴)‘𝑥) = ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥))
16 eldifn 3717 . . . . . . . . 9 (𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴)) → ¬ 𝑥 ∈ dom (𝐹𝐴))
17 ndmfv 6185 . . . . . . . . 9 𝑥 ∈ dom (𝐹𝐴) → ((𝐹𝐴)‘𝑥) = ∅)
1816, 17syl 17 . . . . . . . 8 (𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴)) → ((𝐹𝐴)‘𝑥) = ∅)
1918iuneq2i 4512 . . . . . . 7 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥) = 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))∅
20 iun0 4549 . . . . . . 7 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))∅ = ∅
2119, 20eqtri 2643 . . . . . 6 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥) = ∅
2221uneq2i 3748 . . . . 5 ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥)) = ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ ∅)
23 un0 3945 . . . . 5 ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ ∅) = 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥)
2422, 23eqtri 2643 . . . 4 ( 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥) ∪ 𝑥 ∈ (𝐴 ∖ dom (𝐹𝐴))((𝐹𝐴)‘𝑥)) = 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥)
2515, 24eqtri 2643 . . 3 𝑥 ∈ (dom (𝐹𝐴) ∪ (𝐴 ∖ dom (𝐹𝐴)))((𝐹𝐴)‘𝑥) = 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥)
26 fvres 6174 . . . 4 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
2726iuneq2i 4512 . . 3 𝑥𝐴 ((𝐹𝐴)‘𝑥) = 𝑥𝐴 (𝐹𝑥)
2814, 25, 273eqtr3ri 2652 . 2 𝑥𝐴 (𝐹𝑥) = 𝑥 ∈ dom (𝐹𝐴)((𝐹𝐴)‘𝑥)
29 df-ima 5097 . . 3 (𝐹𝐴) = ran (𝐹𝐴)
3029unieqi 4418 . 2 (𝐹𝐴) = ran (𝐹𝐴)
315, 28, 303eqtr4g 2680 1 (Fun 𝐹 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  wcel 1987  cdif 3557  cun 3558  cin 3559  wss 3560  c0 3897   cuni 4409   ciun 4492  dom cdm 5084  ran crn 5085  cres 5086  cima 5087  Fun wfun 5851   Fn wfn 5852  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-fv 5865
This theorem is referenced by:  funiunfvf  6472  eluniima  6473  marypha2lem4  8304  r1limg  8594  r1elssi  8628  r1elss  8629  ackbij2  9025  r1om  9026  ttukeylem6  9296  isacs2  16254  mreacs  16259  acsfn  16260  isacs5  17112  dprdss  18368  dprd2dlem1  18380  dmdprdsplit2lem  18384  uniioombllem3a  23292  uniioombllem4  23294  uniioombllem5  23295  dyadmbl  23308  mblfinlem1  33117  ovoliunnfl  33122  voliunnfl  33124
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