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Theorem funiunfv 6160
 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 , 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
Distinct variable groups:   ,   ,

Proof of Theorem funiunfv
StepHypRef Expression
1 funres 5632 . . . 4
2 funfn 5622 . . . 4
31, 2sylib 196 . . 3
4 fniunfv 6159 . . 3
53, 4syl 16 . 2
6 undif2 3904 . . . . 5
7 dmres 5299 . . . . . . 7
8 inss1 3717 . . . . . . 7
97, 8eqsstri 3533 . . . . . 6
10 ssequn1 3673 . . . . . 6
119, 10mpbi 208 . . . . 5
126, 11eqtri 2486 . . . 4
13 iuneq1 4344 . . . 4
1412, 13ax-mp 5 . . 3
15 iunxun 4412 . . . 4
16 eldifn 3626 . . . . . . . . 9
17 ndmfv 5895 . . . . . . . . 9
1816, 17syl 16 . . . . . . . 8
1918iuneq2i 4349 . . . . . . 7
20 iun0 4386 . . . . . . 7
2119, 20eqtri 2486 . . . . . 6
2221uneq2i 3654 . . . . 5
23 un0 3810 . . . . 5
2422, 23eqtri 2486 . . . 4
2515, 24eqtri 2486 . . 3
26 fvres 5885 . . . 4
2726iuneq2i 4349 . . 3
2814, 25, 273eqtr3ri 2495 . 2
29 df-ima 5017 . . 3
3029unieqi 4258 . 2
315, 28, 303eqtr4g 2523 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  =wceq 1395  e.wcel 1818  \cdif 3472  u.cun 3473  i^icin 3474  C_wss 3475   c0 3784  U.cuni 4249  U_ciun 4330  domcdm 5004  rancrn 5005  |cres 5006  "cima 5007  Funwfun 5587  Fnwfn 5588  cfv 5593 This theorem is referenced by:  funiunfvf  6161  eluniima  6162  marypha2lem4  7918  r1limg  8210  r1elssi  8244  r1elss  8245  ackbij2  8644  r1om  8645  ttukeylem6  8915  isacs2  15050  mreacs  15055  acsfn  15056  isacs5  15802  dprdss  17076  dprd2dlem1  17090  dmdprdsplit2lem  17094  uniioombllem3a  21993  uniioombllem4  21995  uniioombllem5  21996  dyadmbl  22009  mblfinlem1  30051  ovoliunnfl  30056  voliunnfl  30058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-fv 5601
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