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Theorem onoviun 7016
Description: A variant of onovuni 7015 with indexed unions. (Contributed by Eric Schmidt, 26-May-2009.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
Hypotheses
Ref Expression
onovuni.1  |-  ( Lim  y  ->  ( A F y )  = 
U_ x  e.  y  ( A F x ) )
onovuni.2  |-  ( ( x  e.  On  /\  y  e.  On  /\  x  C_  y )  ->  ( A F x )  C_  ( A F y ) )
Assertion
Ref Expression
onoviun  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  ( A F U_ z  e.  K  L )  = 
U_ z  e.  K  ( A F L ) )
Distinct variable groups:    x, y,
z, A    x, F, y, z    x, K, y, z    x, L, y
Allowed substitution hints:    T( x, y, z)    L( z)

Proof of Theorem onoviun
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 dfiun3g 5245 . . . 4  |-  ( A. z  e.  K  L  e.  On  ->  U_ z  e.  K  L  =  U. ran  ( z  e.  K  |->  L ) )
213ad2ant2 1019 . . 3  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  U_ z  e.  K  L  =  U. ran  ( z  e.  K  |->  L ) )
32oveq2d 6297 . 2  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  ( A F U_ z  e.  K  L )  =  ( A F U. ran  ( z  e.  K  |->  L ) ) )
4 simp1 997 . . . 4  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  K  e.  T )
5 mptexg 6127 . . . 4  |-  ( K  e.  T  ->  (
z  e.  K  |->  L )  e.  _V )
6 rnexg 6717 . . . 4  |-  ( ( z  e.  K  |->  L )  e.  _V  ->  ran  ( z  e.  K  |->  L )  e.  _V )
74, 5, 63syl 20 . . 3  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  ran  (
z  e.  K  |->  L )  e.  _V )
8 simp2 998 . . . . 5  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  A. z  e.  K  L  e.  On )
9 eqid 2443 . . . . . 6  |-  ( z  e.  K  |->  L )  =  ( z  e.  K  |->  L )
109fmpt 6037 . . . . 5  |-  ( A. z  e.  K  L  e.  On  <->  ( z  e.  K  |->  L ) : K --> On )
118, 10sylib 196 . . . 4  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  ( z  e.  K  |->  L ) : K --> On )
12 frn 5727 . . . 4  |-  ( ( z  e.  K  |->  L ) : K --> On  ->  ran  ( z  e.  K  |->  L )  C_  On )
1311, 12syl 16 . . 3  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  ran  (
z  e.  K  |->  L )  C_  On )
14 dmmptg 5494 . . . . . 6  |-  ( A. z  e.  K  L  e.  On  ->  dom  ( z  e.  K  |->  L )  =  K )
15143ad2ant2 1019 . . . . 5  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  dom  (
z  e.  K  |->  L )  =  K )
16 simp3 999 . . . . 5  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  K  =/=  (/) )
1715, 16eqnetrd 2736 . . . 4  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  dom  (
z  e.  K  |->  L )  =/=  (/) )
18 dm0rn0 5209 . . . . 5  |-  ( dom  ( z  e.  K  |->  L )  =  (/)  <->  ran  ( z  e.  K  |->  L )  =  (/) )
1918necon3bii 2711 . . . 4  |-  ( dom  ( z  e.  K  |->  L )  =/=  (/)  <->  ran  ( z  e.  K  |->  L )  =/=  (/) )
2017, 19sylib 196 . . 3  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  ran  (
z  e.  K  |->  L )  =/=  (/) )
21 onovuni.1 . . . 4  |-  ( Lim  y  ->  ( A F y )  = 
U_ x  e.  y  ( A F x ) )
22 onovuni.2 . . . 4  |-  ( ( x  e.  On  /\  y  e.  On  /\  x  C_  y )  ->  ( A F x )  C_  ( A F y ) )
2321, 22onovuni 7015 . . 3  |-  ( ( ran  ( z  e.  K  |->  L )  e. 
_V  /\  ran  ( z  e.  K  |->  L ) 
C_  On  /\  ran  (
z  e.  K  |->  L )  =/=  (/) )  -> 
( A F U. ran  ( z  e.  K  |->  L ) )  = 
U_ x  e.  ran  ( z  e.  K  |->  L ) ( A F x ) )
247, 13, 20, 23syl3anc 1229 . 2  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  ( A F U. ran  (
z  e.  K  |->  L ) )  =  U_ x  e.  ran  ( z  e.  K  |->  L ) ( A F x ) )
25 oveq2 6289 . . . . . . 7  |-  ( x  =  L  ->  ( A F x )  =  ( A F L ) )
2625eleq2d 2513 . . . . . 6  |-  ( x  =  L  ->  (
w  e.  ( A F x )  <->  w  e.  ( A F L ) ) )
279, 26rexrnmpt 6026 . . . . 5  |-  ( A. z  e.  K  L  e.  On  ->  ( E. x  e.  ran  ( z  e.  K  |->  L ) w  e.  ( A F x )  <->  E. z  e.  K  w  e.  ( A F L ) ) )
28273ad2ant2 1019 . . . 4  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  ( E. x  e.  ran  (
z  e.  K  |->  L ) w  e.  ( A F x )  <->  E. z  e.  K  w  e.  ( A F L ) ) )
29 eliun 4320 . . . 4  |-  ( w  e.  U_ x  e. 
ran  ( z  e.  K  |->  L ) ( A F x )  <->  E. x  e.  ran  ( z  e.  K  |->  L ) w  e.  ( A F x ) )
30 eliun 4320 . . . 4  |-  ( w  e.  U_ z  e.  K  ( A F L )  <->  E. z  e.  K  w  e.  ( A F L ) )
3128, 29, 303bitr4g 288 . . 3  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  ( w  e.  U_ x  e. 
ran  ( z  e.  K  |->  L ) ( A F x )  <-> 
w  e.  U_ z  e.  K  ( A F L ) ) )
3231eqrdv 2440 . 2  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  U_ x  e.  ran  ( z  e.  K  |->  L ) ( A F x )  =  U_ z  e.  K  ( A F L ) )
333, 24, 323eqtrd 2488 1  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  ( A F U_ z  e.  K  L )  = 
U_ z  e.  K  ( A F L ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   E.wrex 2794   _Vcvv 3095    C_ wss 3461   (/)c0 3770   U.cuni 4234   U_ciun 4315    |-> cmpt 4495   Oncon0 4868   Lim wlim 4869   dom cdm 4989   ran crn 4990   -->wf 5574  (class class class)co 6281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284
This theorem is referenced by:  oeoalem  7247  oeoelem  7249
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