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Theorem elovmporab1w 6111
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.) (Revised by GG, 26-Jan-2024.)
Hypotheses
Ref Expression
elovmporab1w.o  |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  { z  e.  [_ x  /  m ]_ M  |  ph } )
elovmporab1w.v  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  [_ X  /  m ]_ M  e.  _V )
Assertion
Ref Expression
elovmporab1w  |-  ( 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    x, m, y, z
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 elovmporab1w
StepHypRef Expression
1 elovmporab1w.o . . 3  |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  { z  e.  [_ x  /  m ]_ M  |  ph } )
21elmpocl 6105 . 2  |-  ( Z  e.  ( X O Y )  ->  ( X  e.  _V  /\  Y  e.  _V ) )
31a1i 9 . . . . 5  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  O  =  ( x  e.  _V ,  y  e.  _V  |->  { z  e.  [_ x  /  m ]_ M  |  ph } ) )
4 csbeq1 3083 . . . . . . 7  |-  ( x  =  X  ->  [_ x  /  m ]_ M  = 
[_ X  /  m ]_ M )
54ad2antrl 490 . . . . . 6  |-  ( ( ( X  e.  _V  /\  Y  e.  _V )  /\  ( x  =  X  /\  y  =  Y ) )  ->  [_ x  /  m ]_ M  = 
[_ X  /  m ]_ M )
6 sbceq1a 2995 . . . . . . . 8  |-  ( y  =  Y  ->  ( ph 
<-> 
[. Y  /  y ]. ph ) )
7 sbceq1a 2995 . . . . . . . 8  |-  ( x  =  X  ->  ( [. Y  /  y ]. ph  <->  [. X  /  x ]. [. Y  /  y ]. ph ) )
86, 7sylan9bbr 463 . . . . . . 7  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  [. X  /  x ]. [. Y  / 
y ]. ph ) )
98adantl 277 . . . . . 6  |-  ( ( ( X  e.  _V  /\  Y  e.  _V )  /\  ( x  =  X  /\  y  =  Y ) )  ->  ( ph 
<-> 
[. X  /  x ]. [. Y  /  y ]. ph ) )
105, 9rabeqbidv 2755 . . . . 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 2194 . . . . 5  |-  ( ( ( X  e.  _V  /\  Y  e.  _V )  /\  x  =  X
)  ->  _V  =  _V )
12 simpl 109 . . . . 5  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  X  e.  _V )
13 simpr 110 . . . . 5  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  Y  e.  _V )
14 elovmporab1w.v . . . . . 6  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  [_ X  /  m ]_ M  e.  _V )
15 rabexg 4172 . . . . . 6  |-  ( [_ X  /  m ]_ M  e.  _V  ->  { z  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }  e.  _V )
1614, 15syl 14 . . . . 5  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  { z  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }  e.  _V )
17 nfcv 2336 . . . . . . 7  |-  F/_ x X
1817nfel1 2347 . . . . . 6  |-  F/ x  X  e.  _V
19 nfcv 2336 . . . . . . 7  |-  F/_ x Y
2019nfel1 2347 . . . . . 6  |-  F/ x  Y  e.  _V
2118, 20nfan 1576 . . . . 5  |-  F/ x
( X  e.  _V  /\  Y  e.  _V )
22 nfcv 2336 . . . . . . 7  |-  F/_ y X
2322nfel1 2347 . . . . . 6  |-  F/ y  X  e.  _V
24 nfcv 2336 . . . . . . 7  |-  F/_ y Y
2524nfel1 2347 . . . . . 6  |-  F/ y  Y  e.  _V
2623, 25nfan 1576 . . . . 5  |-  F/ y ( X  e.  _V  /\  Y  e.  _V )
27 nfsbc1v 3004 . . . . . 6  |-  F/ x [. X  /  x ]. [. Y  /  y ]. ph
28 nfcv 2336 . . . . . . 7  |-  F/_ x M
2917, 28nfcsbw 3117 . . . . . 6  |-  F/_ x [_ X  /  m ]_ M
3027, 29nfrabw 2675 . . . . 5  |-  F/_ x { z  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }
31 nfsbc1v 3004 . . . . . . 7  |-  F/ y
[. Y  /  y ]. ph
3222, 31nfsbcw 3115 . . . . . 6  |-  F/ y
[. X  /  x ]. [. Y  /  y ]. ph
33 nfcv 2336 . . . . . . 7  |-  F/_ y M
3422, 33nfcsbw 3117 . . . . . 6  |-  F/_ y [_ X  /  m ]_ M
3532, 34nfrabw 2675 . . . . 5  |-  F/_ y { z  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }
363, 10, 11, 12, 13, 16, 21, 26, 22, 19, 30, 35ovmpodxf 6036 . . . 4  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( X O Y )  =  { z  e.  [_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }
)
3736eleq2d 2263 . . 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 df-3an 982 . . . . 5  |-  ( ( X  e.  _V  /\  Y  e.  _V  /\  Z  e.  [_ X  /  m ]_ M )  <->  ( ( X  e.  _V  /\  Y  e.  _V )  /\  Z  e.  [_ X  /  m ]_ M ) )
3938simplbi2com 1455 . . . 4  |-  ( Z  e.  [_ X  /  m ]_ M  ->  (
( X  e.  _V  /\  Y  e.  _V )  ->  ( X  e.  _V  /\  Y  e.  _V  /\  Z  e.  [_ X  /  m ]_ M ) ) )
40 elrabi 2913 . . . 4  |-  ( Z  e.  { z  e. 
[_ X  /  m ]_ M  |  [. X  /  x ]. [. Y  /  y ]. ph }  ->  Z  e.  [_ X  /  m ]_ M )
4139, 40syl11 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 ) ) )
4237, 41sylbid 150 . 2  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( Z  e.  ( X O Y )  ->  ( X  e. 
_V  /\  Y  e.  _V  /\  Z  e.  [_ X  /  m ]_ M
) ) )
432, 42mpcom 36 1  |-  ( Z  e.  ( X O Y )  ->  ( X  e.  _V  /\  Y  e.  _V  /\  Z  e. 
[_ X  /  m ]_ M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2164   {crab 2476   _Vcvv 2760   [.wsbc 2985   [_csb 3080  (class class class)co 5910    e. cmpo 5912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4565
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-iota 5207  df-fun 5248  df-fv 5254  df-ov 5913  df-oprab 5914  df-mpo 5915
This theorem is referenced by:  elovmpowrd  10945
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