MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elovmpt2rab1 Structured version   Unicode version

Theorem elovmpt2rab1 6530
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 6525 . 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 3404 . . . . . . 7  |-  ( x  =  X  ->  [_ x  /  m ]_ M  = 
[_ X  /  m ]_ M )
54ad2antrl 732 . . . . . 6  |-  ( ( ( X  e.  _V  /\  Y  e.  _V )  /\  ( x  =  X  /\  y  =  Y ) )  ->  [_ x  /  m ]_ M  = 
[_ X  /  m ]_ M )
6 sbceq1a 3316 . . . . . . . 8  |-  ( y  =  Y  ->  ( ph 
<-> 
[. Y  /  y ]. ph ) )
7 sbceq1a 3316 . . . . . . . 8  |-  ( x  =  X  ->  ( [. Y  /  y ]. ph  <->  [. X  /  x ]. [. Y  /  y ]. ph ) )
86, 7sylan9bbr 705 . . . . . . 7  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  [. X  /  x ]. [. Y  / 
y ]. ph ) )
98adantl 467 . . . . . 6  |-  ( ( ( X  e.  _V  /\  Y  e.  _V )  /\  ( x  =  X  /\  y  =  Y ) )  ->  ( ph 
<-> 
[. X  /  x ]. [. Y  /  y ]. ph ) )
105, 9rabeqbidv 3082 . . . . 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 2430 . . . . 5  |-  ( ( ( X  e.  _V  /\  Y  e.  _V )  /\  x  =  X
)  ->  _V  =  _V )
12 simpl 458 . . . . 5  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  X  e.  _V )
13 simpr 462 . . . . 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 4575 . . . . . 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 2591 . . . . . . 7  |-  F/_ x X
1817nfel1 2607 . . . . . 6  |-  F/ x  X  e.  _V
19 nfcv 2591 . . . . . . 7  |-  F/_ x Y
2019nfel1 2607 . . . . . 6  |-  F/ x  Y  e.  _V
2118, 20nfan 1986 . . . . 5  |-  F/ x
( X  e.  _V  /\  Y  e.  _V )
22 nfcv 2591 . . . . . . 7  |-  F/_ y X
2322nfel1 2607 . . . . . 6  |-  F/ y  X  e.  _V
24 nfcv 2591 . . . . . . 7  |-  F/_ y Y
2524nfel1 2607 . . . . . 6  |-  F/ y  Y  e.  _V
2623, 25nfan 1986 . . . . 5  |-  F/ y ( X  e.  _V  /\  Y  e.  _V )
27 nfsbc1v 3325 . . . . . 6  |-  F/ x [. X  /  x ]. [. Y  /  y ]. ph
28 nfcv 2591 . . . . . . 7  |-  F/_ x M
2917, 28nfcsb 3419 . . . . . 6  |-  F/_ x [_ X  /  m ]_ M
3027, 29nfrab 3017 . . . . 5  |-  F/_ x { z  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }
31 nfsbc1v 3325 . . . . . . 7  |-  F/ y
[. Y  /  y ]. ph
3222, 31nfsbc 3327 . . . . . 6  |-  F/ y
[. X  /  x ]. [. Y  /  y ]. ph
33 nfcv 2591 . . . . . . 7  |-  F/_ y M
3422, 33nfcsb 3419 . . . . . 6  |-  F/_ y [_ X  /  m ]_ M
3532, 34nfrab 3017 . . . . 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 6436 . . . 4  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( X O Y )  =  { z  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }
)
3736eleq2d 2499 . . 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 3232 . . . . 5  |-  ( Z  e.  { z  e. 
[_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }  ->  Z  e.  [_ X  /  m ]_ M )
39 df-3an 984 . . . . . 6  |-  ( ( X  e.  _V  /\  Y  e.  _V  /\  Z  e.  [_ X  /  m ]_ M )  <->  ( ( X  e.  _V  /\  Y  e.  _V )  /\  Z  e.  [_ X  /  m ]_ M ) )
4039simplbi2com 631 . . . . 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 32 . . 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 218 . 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 37 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   {crab 2786   _Vcvv 3087   [.wsbc 3305   [_csb 3401  (class class class)co 6305    |-> cmpt2 6307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310
This theorem is referenced by:  elovmpt2wrd  12696
  Copyright terms: Public domain W3C validator