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Theorem onovuni 6802
Description: A variant of onfununi 6801 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 2974 . . . . 5  |-  y  e. 
_V
3 oveq2 6098 . . . . . 6  |-  ( z  =  y  ->  ( A F z )  =  ( A F y ) )
4 eqid 2442 . . . . . 6  |-  ( z  e.  _V  |->  ( A F z ) )  =  ( z  e. 
_V  |->  ( A F z ) )
5 ovex 6115 . . . . . 6  |-  ( A F y )  e. 
_V
63, 4, 5fvmpt 5773 . . . . 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 2974 . . . . . . 7  |-  x  e. 
_V
9 oveq2 6098 . . . . . . . 8  |-  ( z  =  x  ->  ( A F z )  =  ( A F x ) )
10 ovex 6115 . . . . . . . 8  |-  ( A F x )  e. 
_V
119, 4, 10fvmpt 5773 . . . . . . 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 4188 . . . 4  |-  U_ x  e.  y  ( (
z  e.  _V  |->  ( A F z ) ) `  x )  =  U_ x  e.  y  ( A F x )
151, 7, 143eqtr4g 2499 . . 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 3396 . . 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 6801 . 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 6376 . . . 4  |-  ( S  e.  T  ->  U. S  e.  _V )
20 oveq2 6098 . . . . 5  |-  ( z  =  U. S  -> 
( A F z )  =  ( A F U. S ) )
21 ovex 6115 . . . . 5  |-  ( A F U. S )  e.  _V
2220, 4, 21fvmpt 5773 . . . 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 1009 . 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 4188 . . 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 2482 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 965    = wceq 1369    e. wcel 1756    =/= wne 2605   _Vcvv 2971    C_ wss 3327   (/)c0 3636   U.cuni 4090   U_ciun 4170    e. cmpt 4349   Oncon0 4718   Lim wlim 4719   ` cfv 5417  (class class class)co 6090
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-sep 4412  ax-nul 4420  ax-pr 4530  ax-un 6371
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5380  df-fun 5419  df-fv 5425  df-ov 6093
This theorem is referenced by:  onoviun  6803
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