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Theorem onovuni 7011
Description: A variant of onfununi 7010 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 3096 . . . . 5  |-  y  e. 
_V
3 oveq2 6285 . . . . . 6  |-  ( z  =  y  ->  ( A F z )  =  ( A F y ) )
4 eqid 2441 . . . . . 6  |-  ( z  e.  _V  |->  ( A F z ) )  =  ( z  e. 
_V  |->  ( A F z ) )
5 ovex 6305 . . . . . 6  |-  ( A F y )  e. 
_V
63, 4, 5fvmpt 5937 . . . . 5  |-  ( y  e.  _V  ->  (
( z  e.  _V  |->  ( A F z ) ) `  y )  =  ( A F y ) )
72, 6ax-mp 5 . . . 4  |-  ( ( z  e.  _V  |->  ( A F z ) ) `  y )  =  ( A F y )
8 vex 3096 . . . . . . 7  |-  x  e. 
_V
9 oveq2 6285 . . . . . . . 8  |-  ( z  =  x  ->  ( A F z )  =  ( A F x ) )
10 ovex 6305 . . . . . . . 8  |-  ( A F x )  e. 
_V
119, 4, 10fvmpt 5937 . . . . . . 7  |-  ( x  e.  _V  ->  (
( z  e.  _V  |->  ( A F z ) ) `  x )  =  ( A F x ) )
128, 11ax-mp 5 . . . . . 6  |-  ( ( z  e.  _V  |->  ( A F z ) ) `  x )  =  ( A F x )
1312a1i 11 . . . . 5  |-  ( x  e.  y  ->  (
( z  e.  _V  |->  ( A F z ) ) `  x )  =  ( A F x ) )
1413iuneq2i 4330 . . . 4  |-  U_ x  e.  y  ( (
z  e.  _V  |->  ( A F z ) ) `  x )  =  U_ x  e.  y  ( A F x )
151, 7, 143eqtr4g 2507 . . 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 3527 . . 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 7010 . 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 6578 . . . 4  |-  ( S  e.  T  ->  U. S  e.  _V )
20 oveq2 6285 . . . . 5  |-  ( z  =  U. S  -> 
( A F z )  =  ( A F U. S ) )
21 ovex 6305 . . . . 5  |-  ( A F U. S )  e.  _V
2220, 4, 21fvmpt 5937 . . . 4  |-  ( U. S  e.  _V  ->  ( ( z  e.  _V  |->  ( A F z ) ) `  U. S
)  =  ( A F U. S ) )
2319, 22syl 16 . . 3  |-  ( S  e.  T  ->  (
( z  e.  _V  |->  ( A F z ) ) `  U. S
)  =  ( A F U. S ) )
24233ad2ant1 1016 . 2  |-  ( ( S  e.  T  /\  S  C_  On  /\  S  =/=  (/) )  ->  (
( z  e.  _V  |->  ( A F z ) ) `  U. S
)  =  ( A F U. S ) )
2512a1i 11 . . . 4  |-  ( x  e.  S  ->  (
( z  e.  _V  |->  ( A F z ) ) `  x )  =  ( A F x ) )
2625iuneq2i 4330 . . 3  |-  U_ x  e.  S  ( (
z  e.  _V  |->  ( A F z ) ) `  x )  =  U_ x  e.  S  ( A F x )
2726a1i 11 . 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 2490 1  |-  ( ( S  e.  T  /\  S  C_  On  /\  S  =/=  (/) )  ->  ( A F U. S )  =  U_ x  e.  S  ( A F x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 972    = wceq 1381    e. wcel 1802    =/= wne 2636   _Vcvv 3093    C_ wss 3458   (/)c0 3767   U.cuni 4230   U_ciun 4311    |-> cmpt 4491   Oncon0 4864   Lim wlim 4865   ` cfv 5574  (class class class)co 6277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-iota 5537  df-fun 5576  df-fv 5582  df-ov 6280
This theorem is referenced by:  onoviun  7012
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