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Theorem onoviun 7011
Description: A variant of onovuni 7010 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 5253 . . . 4  |-  ( A. z  e.  K  L  e.  On  ->  U_ z  e.  K  L  =  U. ran  ( z  e.  K  |->  L ) )
213ad2ant2 1018 . . 3  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  U_ z  e.  K  L  =  U. ran  ( z  e.  K  |->  L ) )
32oveq2d 6298 . 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 996 . . . 4  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  K  e.  T )
5 mptexg 6128 . . . 4  |-  ( K  e.  T  ->  (
z  e.  K  |->  L )  e.  _V )
6 rnexg 6713 . . . 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 997 . . . . 5  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  A. z  e.  K  L  e.  On )
9 eqid 2467 . . . . . 6  |-  ( z  e.  K  |->  L )  =  ( z  e.  K  |->  L )
109fmpt 6040 . . . . 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 5735 . . . 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 5502 . . . . . 6  |-  ( A. z  e.  K  L  e.  On  ->  dom  ( z  e.  K  |->  L )  =  K )
15143ad2ant2 1018 . . . . 5  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  dom  (
z  e.  K  |->  L )  =  K )
16 simp3 998 . . . . 5  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  K  =/=  (/) )
1715, 16eqnetrd 2760 . . . 4  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  dom  (
z  e.  K  |->  L )  =/=  (/) )
18 dm0rn0 5217 . . . . 5  |-  ( dom  ( z  e.  K  |->  L )  =  (/)  <->  ran  ( z  e.  K  |->  L )  =  (/) )
1918necon3bii 2735 . . . 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 7010 . . 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 1228 . 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 6290 . . . . . . 7  |-  ( x  =  L  ->  ( A F x )  =  ( A F L ) )
2625eleq2d 2537 . . . . . 6  |-  ( x  =  L  ->  (
w  e.  ( A F x )  <->  w  e.  ( A F L ) ) )
279, 26rexrnmpt 6029 . . . . 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 1018 . . . 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 4330 . . . 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 4330 . . . 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 2464 . 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 2512 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113    C_ wss 3476   (/)c0 3785   U.cuni 4245   U_ciun 4325    |-> cmpt 4505   Oncon0 4878   Lim wlim 4879   dom cdm 4999   ran crn 5000   -->wf 5582  (class class class)co 6282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285
This theorem is referenced by:  oeoalem  7242  oeoelem  7244
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