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Theorem ovmpt3rab1 30086
Description: The value of an operation defined by the maps-to notation with a function into a class abstraction as a result. The domain of the function and the base set of the class abstraction may depend on the operands, using implicit substitution. (Contributed by AV, 16-Jul-2018.) (Revised by AV, 16-May-2019.)
Hypotheses
Ref Expression
ovmpt3rab1.o  |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  ( z  e.  M  |->  { a  e.  N  |  ph } ) )
ovmpt3rab1.m  |-  ( ( x  =  X  /\  y  =  Y )  ->  M  =  K )
ovmpt3rab1.n  |-  ( ( x  =  X  /\  y  =  Y )  ->  N  =  L )
ovmpt3rab1.p  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  ps )
)
ovmpt3rab1.x  |-  F/ x ps
ovmpt3rab1.y  |-  F/ y ps
Assertion
Ref Expression
ovmpt3rab1  |-  ( ( X  e.  V  /\  Y  e.  W  /\  K  e.  U )  ->  ( X O Y )  =  ( z  e.  K  |->  { a  e.  L  |  ps } ) )
Distinct variable groups:    x, K, y, z    L, a, x, y    N, a    x, V, y    x, W, y   
x, U, y    X, a, x, y, z    Y, a, x, y, z
Allowed substitution hints:    ph( x, y, z, a)    ps( x, y, z, a)    U( z, a)    K( a)    L( z)    M( x, y, z, a)    N( x, y, z)    O( x, y, z, a)    V( z, a)    W( z, a)

Proof of Theorem ovmpt3rab1
StepHypRef Expression
1 ovmpt3rab1.o . . 3  |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  ( z  e.  M  |->  { a  e.  N  |  ph } ) )
21a1i 11 . 2  |-  ( ( X  e.  V  /\  Y  e.  W  /\  K  e.  U )  ->  O  =  ( x  e.  _V ,  y  e.  _V  |->  ( z  e.  M  |->  { a  e.  N  |  ph } ) ) )
3 ovmpt3rab1.m . . . 4  |-  ( ( x  =  X  /\  y  =  Y )  ->  M  =  K )
4 ovmpt3rab1.n . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  N  =  L )
5 ovmpt3rab1.p . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  ps )
)
64, 5rabeqbidv 2965 . . . 4  |-  ( ( x  =  X  /\  y  =  Y )  ->  { a  e.  N  |  ph }  =  {
a  e.  L  |  ps } )
73, 6mpteq12dv 4367 . . 3  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( z  e.  M  |->  { a  e.  N  |  ph } )  =  ( z  e.  K  |->  { a  e.  L  |  ps } ) )
87adantl 463 . 2  |-  ( ( ( X  e.  V  /\  Y  e.  W  /\  K  e.  U
)  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( z  e.  M  |->  { a  e.  N  |  ph } )  =  ( z  e.  K  |->  { a  e.  L  |  ps } ) )
9 eqidd 2442 . 2  |-  ( ( ( X  e.  V  /\  Y  e.  W  /\  K  e.  U
)  /\  x  =  X )  ->  _V  =  _V )
10 elex 2979 . . 3  |-  ( X  e.  V  ->  X  e.  _V )
11103ad2ant1 1004 . 2  |-  ( ( X  e.  V  /\  Y  e.  W  /\  K  e.  U )  ->  X  e.  _V )
12 elex 2979 . . 3  |-  ( Y  e.  W  ->  Y  e.  _V )
13123ad2ant2 1005 . 2  |-  ( ( X  e.  V  /\  Y  e.  W  /\  K  e.  U )  ->  Y  e.  _V )
14 mptexg 5944 . . 3  |-  ( K  e.  U  ->  (
z  e.  K  |->  { a  e.  L  |  ps } )  e.  _V )
15143ad2ant3 1006 . 2  |-  ( ( X  e.  V  /\  Y  e.  W  /\  K  e.  U )  ->  ( z  e.  K  |->  { a  e.  L  |  ps } )  e. 
_V )
16 nfcv 2577 . . . 4  |-  F/_ x X
17 nfcv 2577 . . . 4  |-  F/_ x V
1816, 17nfel 2585 . . 3  |-  F/ x  X  e.  V
19 nfcv 2577 . . . 4  |-  F/_ x Y
20 nfcv 2577 . . . 4  |-  F/_ x W
2119, 20nfel 2585 . . 3  |-  F/ x  Y  e.  W
22 nfcv 2577 . . . 4  |-  F/_ x K
23 nfcv 2577 . . . 4  |-  F/_ x U
2422, 23nfel 2585 . . 3  |-  F/ x  K  e.  U
2518, 21, 24nf3an 1867 . 2  |-  F/ x
( X  e.  V  /\  Y  e.  W  /\  K  e.  U
)
26 nfcv 2577 . . . 4  |-  F/_ y X
27 nfcv 2577 . . . 4  |-  F/_ y V
2826, 27nfel 2585 . . 3  |-  F/ y  X  e.  V
29 nfcv 2577 . . . 4  |-  F/_ y Y
30 nfcv 2577 . . . 4  |-  F/_ y W
3129, 30nfel 2585 . . 3  |-  F/ y  Y  e.  W
32 nfcv 2577 . . . 4  |-  F/_ y K
33 nfcv 2577 . . . 4  |-  F/_ y U
3432, 33nfel 2585 . . 3  |-  F/ y  K  e.  U
3528, 31, 34nf3an 1867 . 2  |-  F/ y ( X  e.  V  /\  Y  e.  W  /\  K  e.  U
)
36 ovmpt3rab1.x . . . 4  |-  F/ x ps
37 nfcv 2577 . . . 4  |-  F/_ x L
3836, 37nfrab 2900 . . 3  |-  F/_ x { a  e.  L  |  ps }
3922, 38nfmpt 4377 . 2  |-  F/_ x
( z  e.  K  |->  { a  e.  L  |  ps } )
40 ovmpt3rab1.y . . . 4  |-  F/ y ps
41 nfcv 2577 . . . 4  |-  F/_ y L
4240, 41nfrab 2900 . . 3  |-  F/_ y { a  e.  L  |  ps }
4332, 42nfmpt 4377 . 2  |-  F/_ y
( z  e.  K  |->  { a  e.  L  |  ps } )
442, 8, 9, 11, 13, 15, 25, 35, 26, 19, 39, 43ovmpt2dxf 6215 1  |-  ( ( X  e.  V  /\  Y  e.  W  /\  K  e.  U )  ->  ( X O Y )  =  ( z  e.  K  |->  { a  e.  L  |  ps } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364   F/wnf 1594    e. wcel 1761   {crab 2717   _Vcvv 2970    e. cmpt 4347  (class class class)co 6090    e. cmpt2 6092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095
This theorem is referenced by:  ovmpt3rabdm  30087  elovmpt3rab1  30088
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