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Theorem onoviun 6444
Description: A variant of onovuni 6443 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 5010 . . . 4  |-  ( A. z  e.  K  L  e.  On  ->  U_ z  e.  K  L  =  U. ran  ( z  e.  K  |->  L ) )
213ad2ant2 977 . . 3  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  U_ z  e.  K  L  =  U. ran  ( z  e.  K  |->  L ) )
32oveq2d 5958 . 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 955 . . . 4  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  K  e.  T )
5 mptexg 5828 . . . 4  |-  ( K  e.  T  ->  (
z  e.  K  |->  L )  e.  _V )
6 rnexg 5019 . . . 4  |-  ( ( z  e.  K  |->  L )  e.  _V  ->  ran  ( z  e.  K  |->  L )  e.  _V )
74, 5, 63syl 18 . . 3  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  ran  (
z  e.  K  |->  L )  e.  _V )
8 simp2 956 . . . . 5  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  A. z  e.  K  L  e.  On )
9 eqid 2358 . . . . . 6  |-  ( z  e.  K  |->  L )  =  ( z  e.  K  |->  L )
109fmpt 5761 . . . . 5  |-  ( A. z  e.  K  L  e.  On  <->  ( z  e.  K  |->  L ) : K --> On )
118, 10sylib 188 . . . 4  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  ( z  e.  K  |->  L ) : K --> On )
12 frn 5475 . . . 4  |-  ( ( z  e.  K  |->  L ) : K --> On  ->  ran  ( z  e.  K  |->  L )  C_  On )
1311, 12syl 15 . . 3  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  ran  (
z  e.  K  |->  L )  C_  On )
14 dmmptg 5249 . . . . . 6  |-  ( A. z  e.  K  L  e.  On  ->  dom  ( z  e.  K  |->  L )  =  K )
15143ad2ant2 977 . . . . 5  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  dom  (
z  e.  K  |->  L )  =  K )
16 simp3 957 . . . . 5  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  K  =/=  (/) )
1715, 16eqnetrd 2539 . . . 4  |-  ( ( K  e.  T  /\  A. z  e.  K  L  e.  On  /\  K  =/=  (/) )  ->  dom  (
z  e.  K  |->  L )  =/=  (/) )
18 dm0rn0 4974 . . . . 5  |-  ( dom  ( z  e.  K  |->  L )  =  (/)  <->  ran  ( z  e.  K  |->  L )  =  (/) )
1918necon3bii 2553 . . . 4  |-  ( dom  ( z  e.  K  |->  L )  =/=  (/)  <->  ran  ( z  e.  K  |->  L )  =/=  (/) )
2017, 19sylib 188 . . 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 6443 . . 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 1182 . 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 5950 . . . . . . 7  |-  ( x  =  L  ->  ( A F x )  =  ( A F L ) )
2625eleq2d 2425 . . . . . 6  |-  ( x  =  L  ->  (
w  e.  ( A F x )  <->  w  e.  ( A F L ) ) )
279, 26rexrnmpt 5750 . . . . 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 977 . . . 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 3988 . . . 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 3988 . . . 4  |-  ( w  e.  U_ z  e.  K  ( A F L )  <->  E. z  e.  K  w  e.  ( A F L ) )
3128, 29, 303bitr4g 279 . . 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 2356 . 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 2394 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 176    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   E.wrex 2620   _Vcvv 2864    C_ wss 3228   (/)c0 3531   U.cuni 3906   U_ciun 3984    e. cmpt 4156   Oncon0 4471   Lim wlim 4472   dom cdm 4768   ran crn 4769   -->wf 5330  (class class class)co 5942
This theorem is referenced by:  oeoalem  6678  oeoelem  6680
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945
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