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Theorem elovmpt2rab1 6535
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 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.  [_ x  /  m ]_ M  |  ph } ) )
4 csbeq1 3352 . . . . . . 7  |-  ( x  =  X  ->  [_ x  /  m ]_ M  = 
[_ X  /  m ]_ M )
54ad2antrl 742 . . . . . 6  |-  ( ( ( X  e.  _V  /\  Y  e.  _V )  /\  ( x  =  X  /\  y  =  Y ) )  ->  [_ x  /  m ]_ M  = 
[_ X  /  m ]_ M )
6 sbceq1a 3266 . . . . . . . 8  |-  ( y  =  Y  ->  ( ph 
<-> 
[. Y  /  y ]. ph ) )
7 sbceq1a 3266 . . . . . . . 8  |-  ( x  =  X  ->  ( [. Y  /  y ]. ph  <->  [. X  /  x ]. [. Y  /  y ]. ph ) )
86, 7sylan9bbr 715 . . . . . . 7  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  [. X  /  x ]. [. Y  / 
y ]. ph ) )
98adantl 473 . . . . . 6  |-  ( ( ( X  e.  _V  /\  Y  e.  _V )  /\  ( x  =  X  /\  y  =  Y ) )  ->  ( ph 
<-> 
[. X  /  x ]. [. Y  /  y ]. ph ) )
105, 9rabeqbidv 3026 . . . . 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 2472 . . . . 5  |-  ( ( ( X  e.  _V  /\  Y  e.  _V )  /\  x  =  X
)  ->  _V  =  _V )
12 simpl 464 . . . . 5  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  X  e.  _V )
13 simpr 468 . . . . 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 4549 . . . . . 6  |-  ( [_ X  /  m ]_ M  e.  _V  ->  { z  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }  e.  _V )
1614, 15syl 17 . . . . 5  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  { z  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }  e.  _V )
17 nfcv 2612 . . . . . . 7  |-  F/_ x X
1817nfel1 2626 . . . . . 6  |-  F/ x  X  e.  _V
19 nfcv 2612 . . . . . . 7  |-  F/_ x Y
2019nfel1 2626 . . . . . 6  |-  F/ x  Y  e.  _V
2118, 20nfan 2031 . . . . 5  |-  F/ x
( X  e.  _V  /\  Y  e.  _V )
22 nfcv 2612 . . . . . . 7  |-  F/_ y X
2322nfel1 2626 . . . . . 6  |-  F/ y  X  e.  _V
24 nfcv 2612 . . . . . . 7  |-  F/_ y Y
2524nfel1 2626 . . . . . 6  |-  F/ y  Y  e.  _V
2623, 25nfan 2031 . . . . 5  |-  F/ y ( X  e.  _V  /\  Y  e.  _V )
27 nfsbc1v 3275 . . . . . 6  |-  F/ x [. X  /  x ]. [. Y  /  y ]. ph
28 nfcv 2612 . . . . . . 7  |-  F/_ x M
2917, 28nfcsb 3367 . . . . . 6  |-  F/_ x [_ X  /  m ]_ M
3027, 29nfrab 2958 . . . . 5  |-  F/_ x { z  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }
31 nfsbc1v 3275 . . . . . . 7  |-  F/ y
[. Y  /  y ]. ph
3222, 31nfsbc 3277 . . . . . 6  |-  F/ y
[. X  /  x ]. [. Y  /  y ]. ph
33 nfcv 2612 . . . . . . 7  |-  F/_ y M
3422, 33nfcsb 3367 . . . . . 6  |-  F/_ y [_ X  /  m ]_ M
3532, 34nfrab 2958 . . . . 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 6441 . . . 4  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( X O Y )  =  { z  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }
)
3736eleq2d 2534 . . 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 3181 . . . . 5  |-  ( Z  e.  { z  e. 
[_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }  ->  Z  e.  [_ X  /  m ]_ M )
39 df-3an 1009 . . . . . 6  |-  ( ( X  e.  _V  /\  Y  e.  _V  /\  Z  e.  [_ X  /  m ]_ M )  <->  ( ( X  e.  _V  /\  Y  e.  _V )  /\  Z  e.  [_ X  /  m ]_ M ) )
4039simplbi2com 639 . . . . 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 17 . . . 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 223 . 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   {crab 2760   _Vcvv 3031   [.wsbc 3255   [_csb 3349  (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-csb 3350  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:  elovmpt2wrd  12760
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