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Theorem onoviun 6572
Description: A variant of onovuni 6571 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 5089 . . . 4  |-  ( A. z  e.  K  L  e.  On  ->  U_ z  e.  K  L  =  U. ran  ( z  e.  K  |->  L ) )
213ad2ant2 979 . . 3  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  U_ z  e.  K  L  =  U. ran  ( z  e.  K  |->  L ) )
32oveq2d 6064 . 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 957 . . . 4  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  K  e.  T )
5 mptexg 5932 . . . 4  |-  ( K  e.  T  ->  (
z  e.  K  |->  L )  e.  _V )
6 rnexg 5098 . . . 4  |-  ( ( z  e.  K  |->  L )  e.  _V  ->  ran  ( z  e.  K  |->  L )  e.  _V )
74, 5, 63syl 19 . . 3  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  ran  (
z  e.  K  |->  L )  e.  _V )
8 simp2 958 . . . . 5  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  A. z  e.  K  L  e.  On )
9 eqid 2412 . . . . . 6  |-  ( z  e.  K  |->  L )  =  ( z  e.  K  |->  L )
109fmpt 5857 . . . . 5  |-  ( A. z  e.  K  L  e.  On  <->  ( z  e.  K  |->  L ) : K --> On )
118, 10sylib 189 . . . 4  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  ( z  e.  K  |->  L ) : K --> On )
12 frn 5564 . . . 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 5334 . . . . . 6  |-  ( A. z  e.  K  L  e.  On  ->  dom  ( z  e.  K  |->  L )  =  K )
15143ad2ant2 979 . . . . 5  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  dom  (
z  e.  K  |->  L )  =  K )
16 simp3 959 . . . . 5  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  K  =/=  (/) )
1715, 16eqnetrd 2593 . . . 4  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  dom  (
z  e.  K  |->  L )  =/=  (/) )
18 dm0rn0 5053 . . . . 5  |-  ( dom  ( z  e.  K  |->  L )  =  (/)  <->  ran  ( z  e.  K  |->  L )  =  (/) )
1918necon3bii 2607 . . . 4  |-  ( dom  ( z  e.  K  |->  L )  =/=  (/)  <->  ran  ( z  e.  K  |->  L )  =/=  (/) )
2017, 19sylib 189 . . 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 6571 . . 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 1184 . 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 6056 . . . . . . 7  |-  ( x  =  L  ->  ( A F x )  =  ( A F L ) )
2625eleq2d 2479 . . . . . 6  |-  ( x  =  L  ->  (
w  e.  ( A F x )  <->  w  e.  ( A F L ) ) )
279, 26rexrnmpt 5846 . . . . 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 979 . . . 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 4065 . . . 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 4065 . . . 4  |-  ( w  e.  U_ z  e.  K  ( A F L )  <->  E. z  e.  K  w  e.  ( A F L ) )
3128, 29, 303bitr4g 280 . . 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 2410 . 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 2448 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   E.wrex 2675   _Vcvv 2924    C_ wss 3288   (/)c0 3596   U.cuni 3983   U_ciun 4061    e. cmpt 4234   Oncon0 4549   Lim wlim 4550   dom cdm 4845   ran crn 4846   -->wf 5417  (class class class)co 6048
This theorem is referenced by:  oeoalem  6806  oeoelem  6808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051
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