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Theorem elovmpt2rab 6534
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 6530 . 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 3266 . . . . . . . 8  |-  ( y  =  Y  ->  ( ph 
<-> 
[. Y  /  y ]. ph ) )
5 sbceq1a 3266 . . . . . . . 8  |-  ( x  =  X  ->  ( [. Y  /  y ]. ph  <->  [. X  /  x ]. [. Y  /  y ]. ph ) )
64, 5sylan9bbr 715 . . . . . . 7  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  [. X  /  x ]. [. Y  / 
y ]. ph ) )
76adantl 473 . . . . . 6  |-  ( ( ( X  e.  _V  /\  Y  e.  _V )  /\  ( x  =  X  /\  y  =  Y ) )  ->  ( ph 
<-> 
[. X  /  x ]. [. Y  /  y ]. ph ) )
87rabbidv 3022 . . . . 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 2472 . . . . 5  |-  ( ( ( X  e.  _V  /\  Y  e.  _V )  /\  x  =  X
)  ->  _V  =  _V )
10 simpl 464 . . . . 5  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  X  e.  _V )
11 simpr 468 . . . . 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 4549 . . . . . 6  |-  ( M  e.  _V  ->  { z  e.  M  |  [. X  /  x ]. [. Y  /  y ]. ph }  e.  _V )
1412, 13syl 17 . . . . 5  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  { z  e.  M  |  [. X  /  x ]. [. Y  /  y ]. ph }  e.  _V )
15 nfcv 2612 . . . . . . 7  |-  F/_ x X
1615nfel1 2626 . . . . . 6  |-  F/ x  X  e.  _V
17 nfcv 2612 . . . . . . 7  |-  F/_ x Y
1817nfel1 2626 . . . . . 6  |-  F/ x  Y  e.  _V
1916, 18nfan 2031 . . . . 5  |-  F/ x
( X  e.  _V  /\  Y  e.  _V )
20 nfcv 2612 . . . . . . 7  |-  F/_ y X
2120nfel1 2626 . . . . . 6  |-  F/ y  X  e.  _V
22 nfcv 2612 . . . . . . 7  |-  F/_ y Y
2322nfel1 2626 . . . . . 6  |-  F/ y  Y  e.  _V
2421, 23nfan 2031 . . . . 5  |-  F/ y ( X  e.  _V  /\  Y  e.  _V )
25 nfsbc1v 3275 . . . . . 6  |-  F/ x [. X  /  x ]. [. Y  /  y ]. ph
26 nfcv 2612 . . . . . 6  |-  F/_ x M
2725, 26nfrab 2958 . . . . 5  |-  F/_ x { z  e.  M  |  [. X  /  x ]. [. Y  /  y ]. ph }
28 nfsbc1v 3275 . . . . . . 7  |-  F/ y
[. Y  /  y ]. ph
2920, 28nfsbc 3277 . . . . . 6  |-  F/ y
[. X  /  x ]. [. Y  /  y ]. ph
30 nfcv 2612 . . . . . 6  |-  F/_ y M
3129, 30nfrab 2958 . . . . 5  |-  F/_ y { z  e.  M  |  [. X  /  x ]. [. Y  /  y ]. ph }
323, 8, 9, 10, 11, 14, 19, 24, 20, 17, 27, 31ovmpt2dxf 6441 . . . 4  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( X O Y )  =  { z  e.  M  |  [. X  /  x ]. [. Y  /  y ]. ph }
)
3332eleq2d 2534 . . 3  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( Z  e.  ( X O Y )  <-> 
Z  e.  { z  e.  M  |  [. X  /  x ]. [. Y  /  y ]. ph }
) )
34 elrabi 3181 . . . . 5  |-  ( Z  e.  { z  e.  M  |  [. X  /  x ]. [. Y  /  y ]. ph }  ->  Z  e.  M )
35 df-3an 1009 . . . . . 6  |-  ( ( X  e.  _V  /\  Y  e.  _V  /\  Z  e.  M )  <->  ( ( X  e.  _V  /\  Y  e.  _V )  /\  Z  e.  M ) )
3635simplbi2com 639 . . . . 5  |-  ( Z  e.  M  ->  (
( X  e.  _V  /\  Y  e.  _V )  ->  ( X  e.  _V  /\  Y  e.  _V  /\  Z  e.  M )
) )
3734, 36syl 17 . . . 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 223 . 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   {crab 2760   _Vcvv 3031   [.wsbc 3255  (class class class)co 6308    |-> cmpt2 6310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313
This theorem is referenced by: (None)
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