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Theorem onoviun 6819
Description: A variant of onovuni 6818 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 5107 . . . 4  |-  ( A. z  e.  K  L  e.  On  ->  U_ z  e.  K  L  =  U. ran  ( z  e.  K  |->  L ) )
213ad2ant2 1010 . . 3  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  U_ z  e.  K  L  =  U. ran  ( z  e.  K  |->  L ) )
32oveq2d 6122 . 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 988 . . . 4  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  K  e.  T )
5 mptexg 5962 . . . 4  |-  ( K  e.  T  ->  (
z  e.  K  |->  L )  e.  _V )
6 rnexg 6525 . . . 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 989 . . . . 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 5879 . . . . 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 5580 . . . 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 5350 . . . . . 6  |-  ( A. z  e.  K  L  e.  On  ->  dom  ( z  e.  K  |->  L )  =  K )
15143ad2ant2 1010 . . . . 5  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  dom  (
z  e.  K  |->  L )  =  K )
16 simp3 990 . . . . 5  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  K  =/=  (/) )
1715, 16eqnetrd 2641 . . . 4  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  dom  (
z  e.  K  |->  L )  =/=  (/) )
18 dm0rn0 5071 . . . . 5  |-  ( dom  ( z  e.  K  |->  L )  =  (/)  <->  ran  ( z  e.  K  |->  L )  =  (/) )
1918necon3bii 2655 . . . 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 6818 . . 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 1218 . 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 6114 . . . . . . 7  |-  ( x  =  L  ->  ( A F x )  =  ( A F L ) )
2625eleq2d 2510 . . . . . 6  |-  ( x  =  L  ->  (
w  e.  ( A F x )  <->  w  e.  ( A F L ) ) )
279, 26rexrnmpt 5868 . . . . 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 1010 . . . 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 4190 . . . 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 4190 . . . 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 2441 . 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 2479 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 965    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2730   E.wrex 2731   _Vcvv 2987    C_ wss 3343   (/)c0 3652   U.cuni 4106   U_ciun 4186    e. cmpt 4365   Oncon0 4734   Lim wlim 4735   dom cdm 4855   ran crn 4856   -->wf 5429  (class class class)co 6106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-ov 6109
This theorem is referenced by:  oeoalem  7050  oeoelem  7052
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