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Theorem elovmpt2rab1 6504
Description: Implications for the value of an operation, defined by the maps-to notation with a class abstraction as a result, having an element. Here, the base set of the class abstraction depends on the first operand. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Hypotheses
Ref Expression
elovmpt2rab1.o  |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  { z  e.  [_ x  /  m ]_ M  |  ph } )
elovmpt2rab1.v  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  [_ X  /  m ]_ M  e.  _V )
Assertion
Ref Expression
elovmpt2rab1  |-  ( Z  e.  ( X O Y )  ->  ( X  e.  _V  /\  Y  e.  _V  /\  Z  e. 
[_ X  /  m ]_ M ) )
Distinct variable groups:    x, M, y, z    x, X, y, z    x, Y, y, z    z, Z    z, m
Allowed substitution hints:    ph( x, y, z, m)    M( m)    O( x, y, z, m)    X( m)    Y( m)    Z( x, y, m)

Proof of Theorem elovmpt2rab1
StepHypRef Expression
1 elovmpt2rab1.o . . 3  |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  { z  e.  [_ x  /  m ]_ 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.  [_ x  /  m ]_ M  |  ph } ) )
4 csbeq1 3438 . . . . . . 7  |-  ( x  =  X  ->  [_ x  /  m ]_ M  = 
[_ X  /  m ]_ M )
54ad2antrl 727 . . . . . 6  |-  ( ( ( X  e.  _V  /\  Y  e.  _V )  /\  ( x  =  X  /\  y  =  Y ) )  ->  [_ x  /  m ]_ M  = 
[_ X  /  m ]_ M )
6 sbceq1a 3342 . . . . . . . 8  |-  ( y  =  Y  ->  ( ph 
<-> 
[. Y  /  y ]. ph ) )
7 sbceq1a 3342 . . . . . . . 8  |-  ( x  =  X  ->  ( [. Y  /  y ]. ph  <->  [. X  /  x ]. [. Y  /  y ]. ph ) )
86, 7sylan9bbr 700 . . . . . . 7  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  [. X  /  x ]. [. Y  / 
y ]. ph ) )
98adantl 466 . . . . . 6  |-  ( ( ( X  e.  _V  /\  Y  e.  _V )  /\  ( x  =  X  /\  y  =  Y ) )  ->  ( ph 
<-> 
[. X  /  x ]. [. Y  /  y ]. ph ) )
105, 9rabeqbidv 3108 . . . . 5  |-  ( ( ( X  e.  _V  /\  Y  e.  _V )  /\  ( x  =  X  /\  y  =  Y ) )  ->  { z  e.  [_ x  /  m ]_ M  |  ph }  =  { z  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }
)
11 eqidd 2468 . . . . 5  |-  ( ( ( X  e.  _V  /\  Y  e.  _V )  /\  x  =  X
)  ->  _V  =  _V )
12 simpl 457 . . . . 5  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  X  e.  _V )
13 simpr 461 . . . . 5  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  Y  e.  _V )
14 elovmpt2rab1.v . . . . . 6  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  [_ X  /  m ]_ M  e.  _V )
15 rabexg 4597 . . . . . 6  |-  ( [_ X  /  m ]_ M  e.  _V  ->  { z  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }  e.  _V )
1614, 15syl 16 . . . . 5  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  { z  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }  e.  _V )
17 nfcv 2629 . . . . . . 7  |-  F/_ x X
1817nfel1 2645 . . . . . 6  |-  F/ x  X  e.  _V
19 nfcv 2629 . . . . . . 7  |-  F/_ x Y
2019nfel1 2645 . . . . . 6  |-  F/ x  Y  e.  _V
2118, 20nfan 1875 . . . . 5  |-  F/ x
( X  e.  _V  /\  Y  e.  _V )
22 nfcv 2629 . . . . . . 7  |-  F/_ y X
2322nfel1 2645 . . . . . 6  |-  F/ y  X  e.  _V
24 nfcv 2629 . . . . . . 7  |-  F/_ y Y
2524nfel1 2645 . . . . . 6  |-  F/ y  Y  e.  _V
2623, 25nfan 1875 . . . . 5  |-  F/ y ( X  e.  _V  /\  Y  e.  _V )
27 nfsbc1v 3351 . . . . . 6  |-  F/ x [. X  /  x ]. [. Y  /  y ]. ph
28 nfcv 2629 . . . . . . 7  |-  F/_ x M
2917, 28nfcsb 3453 . . . . . 6  |-  F/_ x [_ X  /  m ]_ M
3027, 29nfrab 3043 . . . . 5  |-  F/_ x { z  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }
31 nfsbc1v 3351 . . . . . . 7  |-  F/ y
[. Y  /  y ]. ph
3222, 31nfsbc 3353 . . . . . 6  |-  F/ y
[. X  /  x ]. [. Y  /  y ]. ph
33 nfcv 2629 . . . . . . 7  |-  F/_ y M
3422, 33nfcsb 3453 . . . . . 6  |-  F/_ y [_ X  /  m ]_ M
3532, 34nfrab 3043 . . . . 5  |-  F/_ y { z  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }
363, 10, 11, 12, 13, 16, 21, 26, 22, 19, 30, 35ovmpt2dxf 6410 . . . 4  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( X O Y )  =  { z  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }
)
3736eleq2d 2537 . . 3  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( Z  e.  ( X O Y )  <-> 
Z  e.  { z  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }
) )
38 elrabi 3258 . . . . 5  |-  ( Z  e.  { z  e. 
[_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }  ->  Z  e.  [_ X  /  m ]_ M )
39 df-3an 975 . . . . . 6  |-  ( ( X  e.  _V  /\  Y  e.  _V  /\  Z  e.  [_ X  /  m ]_ M )  <->  ( ( X  e.  _V  /\  Y  e.  _V )  /\  Z  e.  [_ X  /  m ]_ M ) )
4039simplbi2com 627 . . . . 5  |-  ( Z  e.  [_ X  /  m ]_ M  ->  (
( X  e.  _V  /\  Y  e.  _V )  ->  ( X  e.  _V  /\  Y  e.  _V  /\  Z  e.  [_ X  /  m ]_ M ) ) )
4138, 40syl 16 . . . 4  |-  ( Z  e.  { z  e. 
[_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }  ->  ( ( X  e. 
_V  /\  Y  e.  _V )  ->  ( X  e.  _V  /\  Y  e.  _V  /\  Z  e. 
[_ X  /  m ]_ M ) ) )
4241com12 31 . . 3  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( Z  e.  {
z  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }  ->  ( X  e.  _V  /\  Y  e.  _V  /\  Z  e. 
[_ X  /  m ]_ M ) ) )
4337, 42sylbid 215 . 2  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( Z  e.  ( X O Y )  ->  ( X  e. 
_V  /\  Y  e.  _V  /\  Z  e.  [_ X  /  m ]_ M
) ) )
442, 43mpcom 36 1  |-  ( Z  e.  ( X O Y )  ->  ( X  e.  _V  /\  Y  e.  _V  /\  Z  e. 
[_ X  /  m ]_ 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   [_csb 3435  (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-csb 3436  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:  elovmpt2wrd  12544
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