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Theorem onovuni 6359
Description: A variant of onfununi 6358 for operations. (Contributed by Eric Schmidt, 26-May-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
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
onovuni  |-  ( ( S  e.  T  /\  S  C_  On  /\  S  =/=  (/) )  ->  ( A F U. S )  =  U_ x  e.  S  ( A F x ) )
Distinct variable groups:    x, y, A    x, F, y    x, S, y    x, T
Allowed substitution hint:    T( y)

Proof of Theorem onovuni
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 onovuni.1 . . . 4  |-  ( Lim  y  ->  ( A F y )  = 
U_ x  e.  y  ( A F x ) )
2 vex 2791 . . . . 5  |-  y  e. 
_V
3 oveq2 5866 . . . . . 6  |-  ( z  =  y  ->  ( A F z )  =  ( A F y ) )
4 eqid 2283 . . . . . 6  |-  ( z  e.  _V  |->  ( A F z ) )  =  ( z  e. 
_V  |->  ( A F z ) )
5 ovex 5883 . . . . . 6  |-  ( A F y )  e. 
_V
63, 4, 5fvmpt 5602 . . . . 5  |-  ( y  e.  _V  ->  (
( z  e.  _V  |->  ( A F z ) ) `  y )  =  ( A F y ) )
72, 6ax-mp 8 . . . 4  |-  ( ( z  e.  _V  |->  ( A F z ) ) `  y )  =  ( A F y )
8 vex 2791 . . . . . . 7  |-  x  e. 
_V
9 oveq2 5866 . . . . . . . 8  |-  ( z  =  x  ->  ( A F z )  =  ( A F x ) )
10 ovex 5883 . . . . . . . 8  |-  ( A F x )  e. 
_V
119, 4, 10fvmpt 5602 . . . . . . 7  |-  ( x  e.  _V  ->  (
( z  e.  _V  |->  ( A F z ) ) `  x )  =  ( A F x ) )
128, 11ax-mp 8 . . . . . 6  |-  ( ( z  e.  _V  |->  ( A F z ) ) `  x )  =  ( A F x )
1312a1i 10 . . . . 5  |-  ( x  e.  y  ->  (
( z  e.  _V  |->  ( A F z ) ) `  x )  =  ( A F x ) )
1413iuneq2i 3923 . . . 4  |-  U_ x  e.  y  ( (
z  e.  _V  |->  ( A F z ) ) `  x )  =  U_ x  e.  y  ( A F x )
151, 7, 143eqtr4g 2340 . . 3  |-  ( Lim  y  ->  ( (
z  e.  _V  |->  ( A F z ) ) `  y )  =  U_ x  e.  y  ( ( z  e.  _V  |->  ( A F z ) ) `
 x ) )
16 onovuni.2 . . . 4  |-  ( ( x  e.  On  /\  y  e.  On  /\  x  C_  y )  ->  ( A F x )  C_  ( A F y ) )
1716, 12, 73sstr4g 3219 . . 3  |-  ( ( x  e.  On  /\  y  e.  On  /\  x  C_  y )  ->  (
( z  e.  _V  |->  ( A F z ) ) `  x ) 
C_  ( ( z  e.  _V  |->  ( A F z ) ) `
 y ) )
1815, 17onfununi 6358 . 2  |-  ( ( S  e.  T  /\  S  C_  On  /\  S  =/=  (/) )  ->  (
( z  e.  _V  |->  ( A F z ) ) `  U. S
)  =  U_ x  e.  S  ( (
z  e.  _V  |->  ( A F z ) ) `  x ) )
19 uniexg 4517 . . . 4  |-  ( S  e.  T  ->  U. S  e.  _V )
20 oveq2 5866 . . . . 5  |-  ( z  =  U. S  -> 
( A F z )  =  ( A F U. S ) )
21 ovex 5883 . . . . 5  |-  ( A F U. S )  e.  _V
2220, 4, 21fvmpt 5602 . . . 4  |-  ( U. S  e.  _V  ->  ( ( z  e.  _V  |->  ( A F z ) ) `  U. S
)  =  ( A F U. S ) )
2319, 22syl 15 . . 3  |-  ( S  e.  T  ->  (
( z  e.  _V  |->  ( A F z ) ) `  U. S
)  =  ( A F U. S ) )
24233ad2ant1 976 . 2  |-  ( ( S  e.  T  /\  S  C_  On  /\  S  =/=  (/) )  ->  (
( z  e.  _V  |->  ( A F z ) ) `  U. S
)  =  ( A F U. S ) )
2512a1i 10 . . . 4  |-  ( x  e.  S  ->  (
( z  e.  _V  |->  ( A F z ) ) `  x )  =  ( A F x ) )
2625iuneq2i 3923 . . 3  |-  U_ x  e.  S  ( (
z  e.  _V  |->  ( A F z ) ) `  x )  =  U_ x  e.  S  ( A F x )
2726a1i 10 . 2  |-  ( ( S  e.  T  /\  S  C_  On  /\  S  =/=  (/) )  ->  U_ x  e.  S  ( (
z  e.  _V  |->  ( A F z ) ) `  x )  =  U_ x  e.  S  ( A F x ) )
2818, 24, 273eqtr3d 2323 1  |-  ( ( S  e.  T  /\  S  C_  On  /\  S  =/=  (/) )  ->  ( A F U. S )  =  U_ x  e.  S  ( A F x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    C_ wss 3152   (/)c0 3455   U.cuni 3827   U_ciun 3905    e. cmpt 4077   Oncon0 4392   Lim wlim 4393   ` cfv 5255  (class class class)co 5858
This theorem is referenced by:  onoviun  6360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861
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