MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  onovuni Structured version   Unicode version

Theorem onovuni 7005
Description: A variant of onfununi 7004 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 3109 . . . . 5  |-  y  e. 
_V
3 oveq2 6278 . . . . . 6  |-  ( z  =  y  ->  ( A F z )  =  ( A F y ) )
4 eqid 2454 . . . . . 6  |-  ( z  e.  _V  |->  ( A F z ) )  =  ( z  e. 
_V  |->  ( A F z ) )
5 ovex 6298 . . . . . 6  |-  ( A F y )  e. 
_V
63, 4, 5fvmpt 5931 . . . . 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 3109 . . . . . . 7  |-  x  e. 
_V
9 oveq2 6278 . . . . . . . 8  |-  ( z  =  x  ->  ( A F z )  =  ( A F x ) )
10 ovex 6298 . . . . . . . 8  |-  ( A F x )  e. 
_V
119, 4, 10fvmpt 5931 . . . . . . 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 4334 . . . 4  |-  U_ x  e.  y  ( (
z  e.  _V  |->  ( A F z ) ) `  x )  =  U_ x  e.  y  ( A F x )
151, 7, 143eqtr4g 2520 . . 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 3530 . . 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 7004 . 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 6570 . . . 4  |-  ( S  e.  T  ->  U. S  e.  _V )
20 oveq2 6278 . . . . 5  |-  ( z  =  U. S  -> 
( A F z )  =  ( A F U. S ) )
21 ovex 6298 . . . . 5  |-  ( A F U. S )  e.  _V
2220, 4, 21fvmpt 5931 . . . 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 1015 . 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 4334 . . 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 2503 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 971    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106    C_ wss 3461   (/)c0 3783   U.cuni 4235   U_ciun 4315    |-> cmpt 4497   Oncon0 4867   Lim wlim 4868   ` cfv 5570  (class class class)co 6270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273
This theorem is referenced by:  onoviun  7006
  Copyright terms: Public domain W3C validator