MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elovmpt2rab Structured version   Unicode version

Theorem elovmpt2rab 6503
Description: Implications for the value of an operation, defined by the maps-to notation with a class abstraction as a result, having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Hypotheses
Ref Expression
elovmpt2rab.o  |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  { z  e.  M  |  ph } )
elovmpt2rab.v  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  M  e.  _V )
Assertion
Ref Expression
elovmpt2rab  |-  ( Z  e.  ( X O Y )  ->  ( X  e.  _V  /\  Y  e.  _V  /\  Z  e.  M ) )
Distinct variable groups:    x, M, y, z    x, X, y, z    x, Y, y, z    z, Z
Allowed substitution hints:    ph( x, y, z)    O( x, y, z)    Z( x, y)

Proof of Theorem elovmpt2rab
StepHypRef Expression
1 elovmpt2rab.o . . 3  |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  { z  e.  M  |  ph } )
21elmpt2cl 6499 . 2  |-  ( Z  e.  ( X O Y )  ->  ( X  e.  _V  /\  Y  e.  _V ) )
31a1i 11 . . . . 5  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  O  =  ( x  e.  _V ,  y  e.  _V  |->  { z  e.  M  |  ph } ) )
4 sbceq1a 3342 . . . . . . . 8  |-  ( y  =  Y  ->  ( ph 
<-> 
[. Y  /  y ]. ph ) )
5 sbceq1a 3342 . . . . . . . 8  |-  ( x  =  X  ->  ( [. Y  /  y ]. ph  <->  [. X  /  x ]. [. Y  /  y ]. ph ) )
64, 5sylan9bbr 700 . . . . . . 7  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  [. X  /  x ]. [. Y  / 
y ]. ph ) )
76adantl 466 . . . . . 6  |-  ( ( ( X  e.  _V  /\  Y  e.  _V )  /\  ( x  =  X  /\  y  =  Y ) )  ->  ( ph 
<-> 
[. X  /  x ]. [. Y  /  y ]. ph ) )
87rabbidv 3105 . . . . 5  |-  ( ( ( X  e.  _V  /\  Y  e.  _V )  /\  ( x  =  X  /\  y  =  Y ) )  ->  { z  e.  M  |  ph }  =  { z  e.  M  |  [. X  /  x ]. [. Y  /  y ]. ph }
)
9 eqidd 2468 . . . . 5  |-  ( ( ( X  e.  _V  /\  Y  e.  _V )  /\  x  =  X
)  ->  _V  =  _V )
10 simpl 457 . . . . 5  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  X  e.  _V )
11 simpr 461 . . . . 5  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  Y  e.  _V )
12 elovmpt2rab.v . . . . . 6  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  M  e.  _V )
13 rabexg 4597 . . . . . 6  |-  ( M  e.  _V  ->  { z  e.  M  |  [. X  /  x ]. [. Y  /  y ]. ph }  e.  _V )
1412, 13syl 16 . . . . 5  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  { z  e.  M  |  [. X  /  x ]. [. Y  /  y ]. ph }  e.  _V )
15 nfcv 2629 . . . . . . 7  |-  F/_ x X
1615nfel1 2645 . . . . . 6  |-  F/ x  X  e.  _V
17 nfcv 2629 . . . . . . 7  |-  F/_ x Y
1817nfel1 2645 . . . . . 6  |-  F/ x  Y  e.  _V
1916, 18nfan 1875 . . . . 5  |-  F/ x
( X  e.  _V  /\  Y  e.  _V )
20 nfcv 2629 . . . . . . 7  |-  F/_ y X
2120nfel1 2645 . . . . . 6  |-  F/ y  X  e.  _V
22 nfcv 2629 . . . . . . 7  |-  F/_ y Y
2322nfel1 2645 . . . . . 6  |-  F/ y  Y  e.  _V
2421, 23nfan 1875 . . . . 5  |-  F/ y ( X  e.  _V  /\  Y  e.  _V )
25 nfsbc1v 3351 . . . . . 6  |-  F/ x [. X  /  x ]. [. Y  /  y ]. ph
26 nfcv 2629 . . . . . 6  |-  F/_ x M
2725, 26nfrab 3043 . . . . 5  |-  F/_ x { z  e.  M  |  [. X  /  x ]. [. Y  /  y ]. ph }
28 nfsbc1v 3351 . . . . . . 7  |-  F/ y
[. Y  /  y ]. ph
2920, 28nfsbc 3353 . . . . . 6  |-  F/ y
[. X  /  x ]. [. Y  /  y ]. ph
30 nfcv 2629 . . . . . 6  |-  F/_ y M
3129, 30nfrab 3043 . . . . 5  |-  F/_ y { z  e.  M  |  [. X  /  x ]. [. Y  /  y ]. ph }
323, 8, 9, 10, 11, 14, 19, 24, 20, 17, 27, 31ovmpt2dxf 6410 . . . 4  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( X O Y )  =  { z  e.  M  |  [. X  /  x ]. [. Y  /  y ]. ph }
)
3332eleq2d 2537 . . 3  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( Z  e.  ( X O Y )  <-> 
Z  e.  { z  e.  M  |  [. X  /  x ]. [. Y  /  y ]. ph }
) )
34 elrabi 3258 . . . . 5  |-  ( Z  e.  { z  e.  M  |  [. X  /  x ]. [. Y  /  y ]. ph }  ->  Z  e.  M )
35 df-3an 975 . . . . . 6  |-  ( ( X  e.  _V  /\  Y  e.  _V  /\  Z  e.  M )  <->  ( ( X  e.  _V  /\  Y  e.  _V )  /\  Z  e.  M ) )
3635simplbi2com 627 . . . . 5  |-  ( Z  e.  M  ->  (
( X  e.  _V  /\  Y  e.  _V )  ->  ( X  e.  _V  /\  Y  e.  _V  /\  Z  e.  M )
) )
3734, 36syl 16 . . . 4  |-  ( Z  e.  { z  e.  M  |  [. X  /  x ]. [. Y  /  y ]. ph }  ->  ( ( X  e. 
_V  /\  Y  e.  _V )  ->  ( X  e.  _V  /\  Y  e.  _V  /\  Z  e.  M ) ) )
3837com12 31 . . 3  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( Z  e.  {
z  e.  M  |  [. X  /  x ]. [. Y  /  y ]. ph }  ->  ( X  e.  _V  /\  Y  e.  _V  /\  Z  e.  M ) ) )
3933, 38sylbid 215 . 2  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( Z  e.  ( X O Y )  ->  ( X  e. 
_V  /\  Y  e.  _V  /\  Z  e.  M
) ) )
402, 39mpcom 36 1  |-  ( Z  e.  ( X O Y )  ->  ( X  e.  _V  /\  Y  e.  _V  /\  Z  e.  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113   [.wsbc 3331  (class class class)co 6282    |-> cmpt2 6284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator